How to get rid of broken blood vessels on eyelid/around eye skin?

Great question! Eyelid vessels can be treated with a KTP (532nm) laser depending on location and size. I know you couldn't provide a good picture, but it seems like it may be worth to have someone look at the vessels to see if they would be amenable to treatment. You will have to wear metal shields over your eye to protect the eye as the eyelid is treated. A few drops of a numbing solution into your eye will allow painless placement of the eye shield. In treating anywhere around the eye (really anywhere on the body), the eye is most imperative to protect. The angiomas (like red moles) and telangectasia (like red threads) (I presume) will turn a dark color and perhaps appear black as the heat from the laser heats the target and essentially causes the vessel to become damaged and appear dark. This will disappear within a week or two.  

A picture would be helpful but I usually treat these tiny capillaries with a fine needle RF (Ellman) with very excellent results and painless.  Several treatments may be necessary.