How to get a broken torx bit out of a bleed screw? In the latest episode of my disastrous bike maintenance attempts, I got a sub-standard torx bit that broke off in the bleed screw of my Tektro Vela disc brake caliper.

Here's a sligthly overexposed picture of the brake caliper in question, the bleed screw is circled in red:



And another shot from the side. Notice how the bleed port is recessed rather than the bleed nipple you'd usually see on a Tektro brake.



Considering that the screw is slightly recessed into a somewhat expensive safety device, I'd like to find out a way to get the bit out in the least destructive way. If all else fails, I'm considering eating crow and taking my bike to the LBS in the hope that they'll have the tools, experience and/or spare parts to fix this.
From the photo you posted reasonable option is to drill a hole in the bit and use drill extractor, spinning it anti-clockwise.

Bit is made of hardened CrV steel, so drilling it may be hard, since it should be precise and centered. If you feel unsure about your skills, give it to professionals, to locksmith or to your local bike service. It may appear cheaper, since damaging caliper housing would be much more expensive than spending some extra euros to have it fixed.