Hmm... possibly.
You are in denial. Your "possibly" should be "definitely". As you have yourself found...
If I hit F11 for recovery the laptop takes about five minutes to enter the recovery process,
That disk is not even on its last legs. It has passed that stage.
upon which it displays a blue screen ultimately telling me I have a hard disk failure
You saw this with your own eyes.
I cannot access windows.
The disk has failed. Terminally.
If at startup I hit Ctrl-S and go into the BIOS setup and run a Hard Drive Self Test (as suggested by HP - www.hp.com/go/techcenter/support) it results in almost immediate failure and recommends I invest in a new hard drive.
Good advice! The only thing possible.
Is this salvageable?
In my opinion, no.
Only I have a LOT of work on this computer that I need
Sorry to be harsh, but you
had a LOT of work that you didn't back up.
and also my version of Windows (Vista Home Premium) came pre-installed
You may be able to buy a recovery CD from HP that would, in conjunction with a new hard drive, bring your machine back to the state it was in at the time of purchase.
so a disk format is kind of out of the question (if indeed that would even work)
That disk won't take a format, which in any case would make it blank. (I know that when people say "reformat" they generally mean "format the disk and reinstall the operating system")
The only, remote, possibility is that you could try removing the disk and mounting it as a slave in another computer and run some kind of data rescue program just in case anything is recoverable. The cost of this can range from zero or low (a helpful friend) or medium (computer repair shop) to high (specialist data recovery company dismantles the drive under lab conditions and reassembles it with new mechanical and electronic parts in a clean chamber) and there is no guarantee that you would get anything off it.
