You appear to have actually two [& a half] separate issues caused by the same thing.

Firstly the blue spots - which, btw, won't do you any harm so long as you don't stare directly into the light a lot. The bulb appears to have several distinct LEDs inside it - not uncommon at all, & you would probably never notice in normal use, but for this particular purpose, rather a nuisance.

As the lamp runs cool, there's the possibility of an almost free method to do three tasks in one.

Tissue paper. The stuff they used to wrap fresh bread in when you bought it directly from the baker's. It's thin, it's light, it's cheap.

If you add layers of tissue between the lamp & your slide, each layer will diffuse the light more. It will also, incidentally, remove some of the blue cast overall, as it will be slightly off-white.

This brings us to your exposure & white balance.
I assume you have the camera set to the lowest ISO it can - usually 100. If not, do that first. This kind of repetitive task with known input is also one you should have the camera set to fully Manual, so you don't need to keep re-focussing [assuming you're on a fixed tripod, which you should be for this type of task] or resetting exposure etc.

We've already reduced the amount of light coming through by using the tissue paper, & also hopefully reduced the blue cast & bright spots.
All we need to do now is setup up a correct white balance. Do this manually using the tissue-covered lighting you just made.

Check the reference manual for that particular model. Instructions for the D5500 below [click to see larger]