On occasion, when I wanted to remember how many feet there are in a mile (in particular it came up while hiking somewhere, in the pre-mobile-phone days) I've, literally, resorted to the following method:
Remember there are 12 inches in a foot
Remember there are 2.54cm to an inch
work out how many cm in a foot (30.48)
work out how many feet are therefore in a km (km =1000m=100000cm, 100000/30.48 (approximate to 3300)
remember 1km is 5/8 of a mile so get 3300*8/5=5280.
The inch-cm and mile-km conversion factors are far more securely lodged in my head than the number of feet in a mile, so that's the only way I ever seem to be able to remember it.
(A consequence, I suppose, of growing up in a country that started changing to metric in a fit of euro-enthusiasm, and then gave up and stopped half-way - you are constantly having to convert between the two systems...pretty sure I've seen things sold with the length given in meters and the width in inches)
Now I go through it, I'm surprised it comes out right - maybe the errors in the conversion factors happen to cancel out?
The way I think you are supposed to remember it is that there are 8 furlongs in a mile and a furlong is 220 yards i.e.660 feet. But as nobody who doesn't own a racehorse ever thinks about furlongs any more that's not that useful.