I've done something very similar several times now. My kit is now something like 12 years old and the transmitter has randomly died a few times over the years. The humidity reading has also been 99% permanently for the last few years.
I have taken it apart about 2 or 3 times and found similar dead spiders and dense web which I cleaned with water and washing up liquid, and white sediment on the PCB which I scrubbed off with nail varnish remover and a toothbrush. I last did this a few month ago when it died again and ate batteries in a day with the LED just permanently on, but this time I also dried it off with a hairdryer and deliberately got it so hot I couldn't hold it (I used a clamp) but not so hot that things started melting/burning/smelling. I figured the manufacturing process would get hot enough to melt/reflow all the solder which is far hotter than a hairdryer and since it wasn't working, I thought I had nothing to lose.
This seemed to do the trick and the cleaning worked and the heat appeared to bring the humidity sensor back to life too. It's not perfect, in sub zero temperatures it starts flaking out again, but I would certainly recommend the hairdryer blast if anyone has issues with the humidity sensor (without batteries loaded of course!).
Let's hope all of our devices last long into and hopefully beyond 2024!
Mark