You don't need a key puller, TBH. Once you remove the top cover, the keys can be pulled up just with your fingers, by starting from the corner keys and working from there, by grabbing the keys from the opposite sides (between the thumb and the index finger) and pressing into the gap between the key and the base of the keyboard -- it will just pop up. Once all the keys are removed, you can undo the black plastic frame by using a flat screw driver again. Make sure you go as gentle as possible on those plastic loops that you have to push a little sideways to release the notches -- half of them pointing in one direction and the other -- the opposite. Once you release all latches in one direction, the other side is easier.
The rubber dome membrane is under the plastic frame, you can just lift it up -- it's not glued. Then you would have to unscrew and disconnect the PCB, and then you will be able to get to the contact membrane. You can use a multimeter to beep out the leads. Usually, it's only one or two leads, which are not conductive at some point, but because they are serving a few keys, all those keys won't be able to scan by the keyboard processor.
If you are doing it for the first time, I suggest you take pics with your phone as often as you can, to be able to assemble the thing back together. There can be a problem with reinserting the membrane leads back into the PCB connectors (there are different connector designs, some with quick release, and some are not). If the connectors are too tight you can use a piece of thin flat plastic -- I cut these to size (the two terminals are of different size) from the plastic food containers from a grocery store. I pushed the contact ends of the membrane into the terminals by inserting them in between the fold that needs to go in, and then gently pulling just the plastic thing out -- worked like a charm.
Reassembling all the clips are very easy -- they just snap back in, both the frame and the keyboard case. And so do the keys.
If you want to wash the keyboard, while you're there, you can use hot water and dish washing liquid to soak all the keys in for a few minutes, and then give them a quick rub one by one with an old toothbrush. The rubber thing and the black frame, and the case, for that matter (although not the PCB and the membrane matrix) can be all washed under a hose, with some mild dish washing detergent as well. The membrane can be gently wiped with 90% rubbing alcohol, on the outside. The inside layers are not usually dirty at all.
I hope you will be able to fix your keyboard. I was able to fix all the "broken" 401's that I had. And it's much easier than doing so with LK-201, believe me
The screwless design is just genius, IMO.