A "sick bay" tank is an excellent idea. You should change all of the water once or twice a day, depending how much water is in the container.
What I have done is keep a 10 gal tank with dechlorinated water for the water changes, since that way the water has a change to get to the same temperature. I use a silicone colander to scoop the axie up, then dump out the old water, put in the new water, and then put the axie back in. Be very slow the first few time you scoop the axie out, because they can panic and thrash. Usually, once you've done it a few times, it doesn't panic them.
Most likely the unplugged filter is the problem, but if you forgot the dechlorinator, that could have caused the problem also. There are a lot of nasty chemical compounds that build up in a filter without flow (anearobic bacteria - don't ask me for the details, I can't remember.) A big water change with chlorinated water can kill a biofilter. Chloramine (if you have that) is even worse, since it is more stable in water. And if you are rinsing the filter in chlorinated water, that will kill much of your biofilter.
Once you clean the tank, you will need to provide a source of ammonia to get the filter reestablished. Read up on "fishless cycling" - you can used plain household ammonia. Follow the "fishless cycling" direction concerning the ammonia levels - too much ammonia will interfere with the beneficial bacteria. Hopefully, the bio-filter is just stunned, and will bounce back quickly.
Good luck!