I have an office computer chair whose leg broke, there is a huge crack along the plastic, and it bends, but its still attached. I tried using hot glue and duct tape, but it breaks apart again because of the down force twisting the plastic and sheering it.
But if you really want to fix it, you need something to support the weight. Tape and glue aren't going to do it. I would find a length of strong pipe (check the plumbing section at a home improvement store) that will fit nicely either inside the leg (if it is hollow) or over top of it. Make the pipe fill/cover as much of the leg as possible. Then attach with tape, glue, bolts, whatever - just to stop it from sliding out.
If the pipe fits snugly it should hold up fairly well.
Basically it will never be as strong as new.
If you do want to pursue a fix I would suggest rotating (friction) welding with a dremel and plastic filament (preferably identical to the chair).
Or, another idea is to cast epoxy resin or polyester.
I have an identical chair with the exact same break on one of the legs. I attached the broken part to the base by zip ties, and to stop the downward force of my chair possibly breaking it again, i tied a cable to the part where the seat and spine meet to the broken leg
you have to make hole on either side (top hole on broken part ,top hole on stationary part , side wise hole on both side of both part) then take a strong metal strip which contain small hole on both the end. then joint this stationary and broken part with strip and NUT BOLT combination THAT's IS TS's DONE