My employer's laptop has just crapped out - looks like a graphics chip failure.

Things aren't all bad - she has good up-to-date data backups onto an external USB drive, so as soon as a replacement arrives that shouldn't be a problem. Also, as it seems to be a hardware failure in the laptop, I see no reason not to pull the hard drive out of the laptop and stick it in an external enclosure to get any other data out of it.

So, the questions I'm likely to have to address are things like - where are the autocomplete options for IE stored? It will earn me great kudos if I can make IE behave just like before. Similarly, where are cookies saved? If I can get these in place then that will be a result.

Pull HDD out of old laptop
Stick in new laptop
Boot machine
Drink tea
Enjoy kudos

That easy, eh?

I'll ask for beer, not tea

Providing it's the same hardware in the new laptop, the drivers etc should be the same.
Even if it is different, Windows will probably find most of them. Though a clean install in that case would be preferable.
The only thing with just switching the HDD to a new laptop is moving the Windows license. If it's a full one, no problem. If the laptop came with an OEM license, you can't.