How can I run cable through perpendicular studs?
I may understand your situation. I assume it looks like this (in plan view):

 _____
|_____|
   *
   *
   * &lt;-- planned cable
   *
 __*__
|_____|
  | |
**|*|******** &lt;-- existing cable
  |_|


If that's accurate, simply drill a hole at a diagonal, like so, from whichever side gets you the most clearance:

 _____
|_//__|
  | |
  | | 
  |_|


This hole can penetrate the other stud somewhat to keep more centered, and should be safely above or below any nearby cable, both to prevent damage and to allow smooth bends between holes. The hole should be centered on the stud depth on the unobstructed face.

If the cable ends up less than 1-1/4" from the face of any stud (from the perspective of the finished wall surface), add a protective steel plate to the stud edge.

Note that attempting to bend cable in a multi-direction hole through lumber is likely to stress the conductors. All bends should be smooth and of an adequate radius to prevent tearing of the copper wire and insulating jacket. It's also an exercise in frustration.