For a couple of pounds you can get from the chemist a special tool for removing ticks properly, including the mouthparts. Also available online. Left in they will cause infection and an abscess. Because the ticks are close to the eye, you would risk damaging it were you to use some of the old tricks like surgical spirit which sometimes make them reverse out, or touching with a cigarette end which should have the same effect. The special tool is by far the best thing to use, and you should also carry one in your own first aid kit for ticks on people.
You would need to wrap him in a giant bath towel, like you would wrap a cat for giving worm pills, with his feet, wings and neck controlled, preferably with a helper.