Whenever i get my boat going again im going to get one of those digital thermometers with an alarm. You can set those to let you know when the livewell water temp is out of sync. Otherwise, id probably not do it properly myself.
Cut, i've read several responses at sites about cleaning a livewell. Anything goes really, including a bleach/water solution(1:9), saltwater sitting in there/sloshing around, tap water running/pumping out for an hour, baking soda wash/rinse or mild to moderate cleaners of all sorts. Just anything that people could come up with and at least half way be safe about it. Just make sure you rinse thoroughly before putting fish in there as an extra precaution. The bleach solution is highly suggested in helping fight against LMBV.
If you fear that your livewell has indeed been contaminated, by all means clean that rascal. You might want to check the transom area at the motor(while on the water) for a residue or film on the water, some LW intakes are real close and can suck in bad water sometimes. If you have a leak of some sort back there(motor/foot seepage, trim leaking/seeping, etc.), fix the problem and see if it gets better. I've experienced that very thing and is why i'm suggesting you looksee.