With the disclaimer that my turntable stands, these days, unused...

Microfibre Cloth. This is the best thing in the world for cleaning almost anything, and is absolutely wonderful for cleaning LPs. Don't give $£$ for vinyl gadgets made from exotic materials, give $-a-couple-of to your local household store for microfibre. It is as exotic as cleaning materials ever need to get (for everything, from car to bath) and only looks as if it isn't because it is cheap. Although, stuff like e-cloths (which I think were maybe the first makers of microfibre in UK) are actually not cheap.

I'm not saying that one can buy and use blind, especially with the very cheap stuff. It needs to have a pile, to get into those grooves, but it needs to not shed its pile. Try cleaning some glass to check this.

I'm also not saying that it rivals mechanical/fluid cleaners. I know that, when I used to make jewellery on a hobby basis, one of my dream tools was an ultrasonic cleaner, but I never had one (these days they can be had in cheep home versions). I'm sure they work wonders on LPs.