Q:

What size should sistered joists be when repairing a cut joist?

I need to repair a cut joist in my kitchen floor. I was going to use a header and sister the joists on either side of it. Question: how do I know what thickness of a header I should use? What about the thickness of the sistered joists? The joists are 16" on centre. 
Here is a drawing of the problem area 
(for context, this is a top-down view of the floor under the kitchen) :

Here is my proposed fix

A:

If we're talking about any sort of engineered joist (truss, TJI), stop reading now and consult a local engineer. This answer assumes solid 2x10 lumber.

I'm guessing that you don't mean a header so much as a joist fit between the sistered joists alongside. Since the span is so short (presumably 32" or less), you can use a single joist of the same height as the one to be repaired (so you have full backing for joist hangers). 
The sistered joists should also be the same height as the originals. They would not need to have bearing at the ends if they're very well fastened to the original joists (construction adhesive along with 16d nails or structural screws in sets of three at 12" intervals, say). 
|  |  |            |  |             |  |  |
|  |  |            |  |             |  |  |
|  |  |_           |  |           _ |  |  |
|  |  |___________||__||____________|  |  |
|  |  |_____________________________|  |  |
|  |  |_          ||  ||          _ |  |  |
|  |  |            |  |             |  |  |
|  |  |            |  |^- hangers-^ |  |  |
|  |  |            |  |             |  |  | <-- original joist
|  |  |            |  |
|  |  |            |  |  <-- cut joist
|  |  |   
|  |  | <-- sistered joist    

It's best to fit all joist components first (with a few toenails in the case of the cross-joist), then install the hangers. This ensures that they're pressed up tight, and that you can get things in place at all. Be sure to use construction adhesive at all points of contact (wood-to-wood and metal-to-wood) to prevent squeaks. 

