It would theoretically be possible for a technician to repair (or, more likely, replace) the power supply unit without the hard drive being present but it complicates testing and I simply wouldn't bother.
A new keyboard will cost £10 to £15. A replacement PSU could cost
at least £60. The labour charge is likely to be £50. (That's PC World's flat-rate charge for laptop repairs. An independent repairer would probably charge around the same because PSU repairs can be fiddly and time-consuming).
Given those costs, I'd buy a decent secondhand laptop instead. e.g.
http://www.accomputerwarehouse.com/product-info/cheap-windows-7-laptop-hp-6930p-core-2-duo-2gb-ram-14-inch-display
I'd then remove the hard drive from the old laptop. (It's really easy. There are usually just a few screws). Then I'd read the label on it to see whether it's an IDE or SATA drive. (If it's a fairly modern laptop, it'll be SATA. If in doubt, Google the model number).
Next, for a few pounds, I'd buy either an IDE caddy
http://www.ebay.co.uk/itm/like/360591634628?lpid=122&chn=ps&adgroupid=33676502101&rlsatarget=pla-270397376043&adtype=pla&poi=&googleloc=9045888&device=c&campaignid=707291931&crdt=0
or a SATA one
http://www.ebay.co.uk/itm/like/232141462447?lpid=122&chn=ps&adgroupid=36084540643&rlsatarget=pla-272157487575&adtype=pla&poi=&googleloc=9045888&device=c&campaignid=698615505&crdt=0
as appropriate.
It's dead easy to fit the old drive into the caddy. The old drive can then be seen as an external USB drive by the new laptop. Initially, all of your personal files will seem to have vanished but they will magically reappear once you 'take ownership' of the drive by following these simple instructions:
http://preyerplanning.com/take-ownership-of-entire-hard-drive-in-windows-7.pdf
You can then copy your files across to your new laptop and either securely wipe the old one or simply destroy it.
My advice isn't simply theory. It's exactly what I've done myself on several occasions when laptops have packed up.