Wow, well done getting that far! IT's quite unnevering to do, but it should all be fixable. Don't reassemble it just yet - best to test it first.
For testing it - with the plastic keyboard part removed, you can now reconnect the keyboard and test out which keys are working. I expect that just cleaning it won't be sufficient and you'll need to reconnect some traces using conductive paint. I watched a couple of youtube videos for membrane keyboard repair. This one explains the process really well: https://www.youtube.com/watch?v=REz928pgCII
This is also a good video: https://www.youtube.com/watch?v=7i5D8xPsGCk
To repair the traces, you just need something that's conductive and adhesive. Three methods:
1. Conductive paint applied with a syringe or needle tip
2. Using Kaptop tape, stick a single strand of copper (remove from a cut mains cable or something) over the broken point, following the trace. I just spotted this in this video
3. IF you have some copper slug tape (which you can buy in lots of stores), cut off a piece of that and shape it like the trace but a bit too long. Adhere the middle part over the broken trace part, then fold over the ends so that the copper touches at each end, then stick down with kapton tape (if you can't tell, I like kapton/polyamide tape - it's more permanent than sellotape, is quite thin and a good insulator).
There's an even better way - instead of using slug tape, use actual EMI tape, which is adhesive and conductive on both the copper and adhesive sides, then you just need to cut something in the shape of the trace and stick it down.
It took me a while to find the fault on my Toshiba's keyboard, but on both my bad keyboards, it was near the edge since that's where water ingress can most easily occur. For the Toshiba keyboard, I took a picture of both sides, mapped out which keys were bad and used that to draw lines and deduce where the fault was.
But mostly, just inspect carefully under a bright light! In both cases, the faulty part was a slightly discoloured / darkened part of the trace.
When it does come to reassembling, use some non-serrated needle pliers, or some tweezers, to carefully squeeze the plastic posts to match up with the flat parts of the holes in the aluminium. That will usually allow the pegs to go back in and increase the height / plastic amount enough that heat staking them back down should work. IF you have a larger tip for the soldering iron, I found that the larger area is easier to squash down the plastic, like the last picture of my 730CDT keyboard repair.
Also I gave you the wrong temperature for peg removal, sorry - I used 200C for my better repair, that will melt slowly allowing the plastic to deform in a more orderly fashion than it will with the higher temp. 200 or 210C should work fine for reassembly.