Calling all chemistry experts .. "there's something to discover here" .. OK .. the situation : non-stick pan AND grease ( oil , butter, animal fat, anything ). WHY is *SO HARD* to remove it from that ? It's like the detergent ( Fairy ) does not "react" and/or "reacts poorly" with it, why it takes so much effort to clean/rinse it/make it non-greasy again ? I'd swear "something unusual is happening here" and I believe the non-stick coating has something to do with it.
@gilesgoat Chemistry expert here: is it just grease? Hot water (plenty) soap and time will take it away.
Is it very greasy and you want to quickly clean it out? Use some powder to absorb it and then throw it away, I use used coffee powder from the moka, but I guess flour or stuff like that shall work as well.
I don't know what fairy contains, for the dishes I generally just use Marseille soap which works and it's cheap. If you want next level degreasing find yourself some sodium laurylsulphate and you'll be stripping grease from anywhere.
But also: if you're cooking with plenty of fat just don't use a non stick pan, you don't need it.
