Here are your parts
Replacement parts for WHIRLPOOL KSRD25FKSS04 | AppliancePartsPros.com

The coil is Item 22 in section 12

I recently had the compresser replaced on this 6 year old unit $400. 2 months later the fridge was warm again.
If the compressor was running and you just did not have cooling originally then the repairman may have replaced the wrong part to start with.

I had the same repairman come out and he informed me the evaporater was leaking freon and needed to be replaced.
I would want to see the leak. This will confirm that he installed the compressor correctly and the leak is not at one of his connections. Depending where the leak is it may be repairable.

My question is what parts do I need to replace and is this something I can do myself?
Cannot say whet parts you need but it is not a job for a DIYer as you need special equipment and in some areas you have to be certified to work with freon. The repair will not be cheap!!!

I do not like dissing a repairman so I guess it comes down to how much you trust him.

Here is a document that goes over what is involved
https://secured.whirlpoo...63be04/$FILE/4321963.pdf