Your story is tragicomic, unfortunately. I see the funny side but can also see the tragic side. However, it's good thing you can laugh about it. I bet it's not the first time it happens to you. If you have a chronic illness and you're in and out of hospitals, you can't avoid this to happen. Doctors and nurses are not all the same, some are very good with needles, you don't feel a thing when they are taking blood, others can destroy your vein in a second.
I've seen a very unfortunate scene when I was in hospital a few years ago. A nurse tried to administrate antibiotics to an old lady through the cannula in her hand, the old lady told the nurse it hurts like hell, she didn't believe her, which resulted in the old lady having a very swollen hand. The doctor was called in, it was a huge scandal but the damage was done.
I also know that there are rules when it comes to blood drawing. My mom had pancreatic cancer, which meant one and a half year of chemo, so I have a little bit of experience, unfortunately.
So I know that if the nurse failed to take blood, the second time must be chosen a spot lower than the first, I mean father from the heart. This means you lowers arm or hand. I know nurses and doctors are human too and can make mistakes but chemo patients have their veins already destroyed, therefore there is no place for mistakes.
Anyway. I hope your arm is healed and not blue,green and yellow from the incident and wish this won't happen again, although there are no guarantees :) Get healthy soon :)