Why is Coleridge needed in Dirk Gently's Holistic Detective Agency?

The ghost (really an alien) was destroyed after making a terrible mistake. Several billion years later, it realised that Cambridge University presented a unique opportunity, featuring both an individual that it could communicate with (Coleridge) and an individual with access to a time machine (Reg). Frustratingly for the ghost, Coleridge was only able to be influenced when he was deeply relaxado after consuming heroic amounts of laudanum, and hence unable to be a truly useful conduit for its desires.

“I tried to tell him my story,” admitted the ghost, “I—”
“Sorry,” said Dirk, “you’ll have to excuse me—I’ve never cross—examined a four-billion-year-old ghost before. Are we talking Samuel Taylor here? Are you saying you told your story to Samuel Taylor Coleridge?”
“I was able to enter his mind at… certain times. When he was in an impressionable state.”
“You mean when he was on laudanum?” said Richard.
“That is correct. He was more relaxed then.”
“I’ll say,” snorted Reg, “I sometimes encountered him when he was quite astoundingly relaxed. Look, I’ll make some coffee.”
He disappeared into the kitchen, where he could be heard laughing to himself. “It’s another world,” muttered Richard to himself, sitting down and shaking his head.
“But unfortunately when he was fully in possession of himself I, so to speak, was not,” said the ghost, “and so that failed. And what he wrote was very garbled.”

In an attempt to make Coleridge more amenable to helping (possibly to gain sympathy?) the ghost evidently recounted its story, which Coleridge later turned into a poem. On several occasions, the ghost was able to gain sufficient control over Coleridge to actually get him to speak to Reg, but it was again frustrated to find that very thing that made him receptive (the laudanum) prevented him from making any sort of useful contact with the Professor.

“Professor,” called out Dirk, “this may sound absurd. Did—Coleridge ever try to… er… use your time machine? Feel free to discuss the question in any way which appeals to you.”
“Well, do you know,” said Reg, looking round the door, “he did come in prying around on one occasion, but I think he was in a great deal too relaxed a state to do anything.”

Spin forward two hundred years and the ghost's story (as related in the poetry of Coleridge) was successful in influencing Michael Wenton-Weakes to become the perfect conduit for its will.

The words were very familiar to him, and yet as he read on through them they awoke in him strange sensations and fearful memories that he knew were not his. There reared up inside him a sense of loss and desolation of terrifying intensity which, while he knew it was not his own, resonated so perfectly now with his own aggrievements that he could not but surrender to it absolutely.

When Dirk changes the poem (by distracting Coleridge) the ultimate effect is to prevent Michael from being influenced, and hence leaving the ghost impotent to travel back in time and prevent its ship from exploding.

As an additional point in response to the original question regarding the significance of Coleridge's involvement, apparently Coleridge's completed poem (in the initial timeline) contained encoded technical details regarding how to fix the space ship.

The reason the alien ghost wanted these details in the poem was to make sure he (it?) would recall exactly how to fix the crippled ship. In a fairly long conversation, Reg explains how his own advanced age and the limited capacity of his brain has caused him to forget much of what he knew in the past. When you read this, you wonder, why is Adams spending so much time explaining how a memory can fade and disappear with time?

Well, since everything is connected in the story, what Adams is suggesting is that the ghost alien also feared that after billions of years he could lose the memory of the engineering details of how to fix his own ship. Indeed, he must have been frantically searching through time for a person who could transcribe these technical details in some form that could be used to complete the repair when the need arose.

So Coleridge was the conduit by which the alien stored his mental spaceship repair manual. While it was fortuitous that the completed Coleridge poem reinforced the ghost's control over Michael Wenton-Weakes, there was no way the ghost could have planned this at the time he dumped the data for the poem into Coleridge's mind. He just wanted to store the repair manual. Apparently once he forced Coleridge to record the repair manual, the ghost could relax his mind and not strain to retain the technical details; so the ghost's memory of the repair information faded over two hundred years after Coleridge's poem was written. By the time the ghost took over Wenton-Weakes, the completed poem was required to implement the spaceship repair.

When Dirk, as the Person from Porlock, went back in time and disrupted Coleridge's memory of the technical details embedded in the poem, a significant portion of the poem disappeared from the new time line and left the ghost with only his faded memory to complete the repair, even assuming that he was able to control Weakes by just getting him to kill the editor who had taken over Weakes' position at a magazine.

In the new time line events proceeded much as they did in the initial time line, with Reg traveling in time to please a little girl and inadvertently opening the door to the Electric Monk who then comes to earth and kills Susan's brother. Apparently the dead brother's experiences as a ghost are provided as a literary device to convince the reader that ghosts do exist and therefore make it easier to accept that an alien ghost can roam the earth for eons.

Anyway, in the new time line Weakes kills the hated replacement editor and is taken over by the alien ghost, but without reference to any completed Coleridge poem. The ghost then convinces Dirk and Reg to take him back in time where, without the completed Coleridge poem, he is unable to make the required repair and again blows up the ship and kick-starts life on earth; except now you have the original alien ghost and the alien ghost from Weakes' body roaming the earth and no doubt plotting a way to kill mankind in yet another timeline. Alternatively, if Weakes can't be taken over in the new timeline without the complete Coleridge poem, then his replacement editor is spared and there remains only the original alien ghost lurking in the shadows, waiting for a chance to somehow save his ship and extinguish mankind.

Of course, the alien ghost's hope for a re-do of the timeline is dashed when the time machine is disabled due to a phone repair! (I didn't see that coming). Also, Reg sneaks back into the alien ship and plants some explosives which blow it up with a flashy light display in the night sky. Reg also brings back some of the weird beautiful alien music from the ship, goes back in time and gives it to Bach and makes Bach a music legend in the new timeline. Reg also returns the Electric Monk and his horse to their home planet just before the time machine is destroyed.

On the subject of the time machine, there is no information other than it uses a highly advanced computer well beyond human technology, with no explanation regarding how Reg obtained it. Apparently, Adams felt the origin of the time machine was not worth discussing. We know that the idea of a time machine with a little door and a big interior was lifted from Adams' work on Doctor Who. He probably didn't want to spend time developing a rationale for such a machine in Reg's universe and therefore decided to treat Reg as a kind of addled time lord and moved on.

In the new timeline for some unexplained reason the alteration of Coleridge's poem results in the lost cat of the old time line living and dying in the new timeline without ever being lost. Of course, Dirk takes credit for this and for saving the human race.

Finally, what is the point of introducing a sofa that is wedged in a stairwell by some means that transcends the laws of physics? This is a tough one to figure out. What could Adams have been thinking?

I think the stuck sofa is like the complete Coleridge poem. Like the sofa, the complete poem is an anomalous, even paradoxical thing existing in the tangled web of time with the potential to block forward progress. Thus, the complete poem is wedged in place and can be used to block the evolution of all earth life. How can it be eliminated?

The only way, as with the sofa, is to cut it up and remove it. By becoming the Person from Porlock, Dirk cuts up the poem and eliminates it as an obstacle, just like the police did when they sawed up the sofa and carried its pieces away, leaving only a pile of sawdust. Dirk eliminates the second detailed part of the poem, leaving only the useless "sawdust" of the first portion which cannot by itself block the evolution of humanity. I know this is pretty far out, but it's the best I can do to try to explain how the stuck sofa is connected to the solution of the Coleridge poem problem. So everything is connected, even the sofa and the poem.

This convoluted explanation is provided in an effort to understand at least some of the twisted logic of the first Dirk Gently book. I hope Adams is smiling, wherever he is.