I was taking a nap and when I woke up I had a broken blood vessel on my eyelid, can it be fixed? (Photo)

Sorry to hear about that. Could be just accidental aggressive rub while sleeping but if its a ruptured blood vessel it should heal on its own in couple of weeks.

Laser therapy has evolved to address small and medium sized abnormal veins. In fact, endovenous laser therapy is useful for the management of varicose-vein associated venous insufficiency. Sclerotherapy using foam sclerosants are ideal for the treatment of minimally-refluxing varicosities and reticular veins. In our office, we combine these less-invasive modalities with microphlebectomy and vein surgery in order to minimize scarring and remove unsightly varicosities. 
Dr. Karamanoukian
Realself100 Surgeon

You should not attempt to have anyone do anything to that area. Sometimes blood vessels spontaneously burst or burst after a sneeze or cough (due to pressure). When this happens in the white area of the eye, it can look quite dramatic, but it is completely harmless. Ignore it and it will go away quickly. You can use a warm compress to speed healing if you like.

Most likely you somehow irritated the vein while sleeping/napping and this caused it to rupture and bleed.  This happen commonly and the best treatment is to leave it alone and it will heal over about 7 to 10 days.