You won't get much simpler than this.
simulate this circuit – Schematic created using CircuitLab
As the temperature rises, the voltage on IN+ risis; at 37C it crosses the threshold and the LED switches on.
You posted the data for 5 thermistors, I'm assuming the correct one is the right and column based on the numbering (103) where 37C corresponds to 6.2k.
So I made all the other resistors in the bridge the same value, so that they are likely to be from the same batch, reducing error : 1% tolerance is probably good enough, go for 0.5% if you can afford it. Supply is not critical, a 9V battery is OK but 5V should work too. Or 2 Lithium coin cells, a bit smaller.
It appears that assumption is wrong; for the thermistor labelled "10K" the correct resistance is 6.016K. Simplest fix is to bring R4 down to (about) 6.016K by adding 200K in parallel with it, so that's what I have done. (6K resistors are a bit less readily available).
Adjust R5 for suitable brightness.
The LM358 is a pretty standard opamp (datasheet here) - you probably want the LM358P or another variant with the DIP-8 package. On Digikey, eliminate "surface mount" from the search and it's easy to find.
And keep washing your hands.