help determining rim size
 

hey all so i found an old basket bike in a field one day
im taking the time to fix her up and get her back into good shape
the problem is this is my first bike repair job and i know little about bikes

the bike is in good shape but the rims are seriously rusted out along with the spokes (most are cut or bent). i want to score some new rims for her but i have no clue what to measure for sizing purposes

the basics are 36 spokes, the tires are 26X2.125 inches, the measurements i have done on the rims
are the diameter i got 22 5/8th's by 1 7/16th wide these are rather rough sizes because the rims are in horrible shape (one is crazy bent)

what exactly do i measure for the rim and how do i detemrine the size of the rim

i also want to add a rear brake and the current rims are smooth meaning no place to add brake shoes to grab onto a flat area, but since i am buying new ones this should not be a problem right ( rim is () not || )

any thoughts or advice or even where to look for bike rims would rock
im currently looking at bikepartsusa.com

If you look on the tire, you will see a number like 38/559 or 21/622, something like this marked in the rubber. The bigger number id the rim diameter.

If the tire is 26.125" and not 26 1/8", then it is a 559mm diametre mountain bike rim.