reinforcing existing header
this is called a knee brace
You mean the span not the bearing width. Assuming the brace to be fully effective, the effective span is less than the span between the knee braces. The reason is too complected to explain in detail here. For design, you could be conservative and just use the distance between the braces
I usually suggest lag screws. The screws will be in shear, not withdrawal, so the length only needs to be long enough to get through the brace and into the main member a few inches


This isn't a very good solution. You'll create bending in your trimmer and king stud that will reduce their design strength. Knee braces generally aren't used to reduce the span, they are for lateral (sideways) strength so that it doesn't fall over. Think about a gazebo

Your best solution is to replace the header with a stronger one. It can be of the same(ish) size and therefore not reducing the opening under it. Good options include glue laminated beam and manufactured lumber such as LSL, LVL and PSL.