How can I easily fix or replace the broken knob handle on bathtub faucet control?
Depending on the manufacturer, the stem should be a free replacement.
It's fairly trivial to replace. 


Turn off water. (IMPORTANT!)
Remove the large hex fitting on the face of the controller
With vise grips, pull and wiggle the stem out.


If it's a common manufacturer, you should be able to exchange the stem at your local big box store or plumbing supplier.

Reinstalling is simply the reverse.

While picking up the stem, get an appropriate bolt to fit, and you're good to go.

Caveat:  Figure out the manufacturer and call first.