I'm not sure what exactly you're looking for, but I'll do my best.
I won't give you a cheating way because 1) it's not helpful and 2) I'm a professional music teacher and it's against my credo. :)
There is a way to find a minor key signature that works for every major key. Let's take your first example, the key of E minor. In order to find its signature, you need to go a minor third UP, or 3 half-steps. So starting on an "E," a minor third (or 3 half-steps) up is G. The key signature of G Major is.... 1 sharp, right? So, the key signature of E minor is the same, 1 sharp. And since sharps are always in the same order (right?....) it would have to be F sharp. So, the key signature of E minor is one sharp, F#.
Make sense?
This works with any key. For example, a minor third (or three half-steps) up from F is A flat. A flat has 4 flats, and so F minor's key signature is 4 flats (BEAD).
Your last one involves a lot of flats, but the principle is no different. Three half-steps up from Eb is Gb. The key signature for Gb major is a whopping 6 flats, and so it's the same for Eb minor. (BEADGC)