Late 2011 MacBook Pro 15" broken graphics - want to use external monitor
The linked forum thread has 109 pages; What is your current state of MacBook - what have you tried from that forum post? 

If your MacBook is now working and using the integrated graphics card, you can try to buy a USB to VGA/HDMI/... adapter. 

Streaming to an Apple TV (which then connects to a TV/monitor) should work, even without the dedicated graphics card, but there are some limitations to it (e.g. may be slow, not ideal for videos/transitions,...). At least I could not find anything that implied it being a problem.

AirPlay is as far as I know only for music.

But you should probably read more about this problem here.