After I alerted the vet, she confirmed that the vein was blown and she had injected it outside the vein, in the leg. The solution is ph balanced, so that it isn't painful when injected into the vein. It does burn when injected into tissue. That's just the way it physically is. Not being in the vein was also why it wasn't putting her to sleep. It was sitting in the leg, instead of circulating in the bloodstream and being absorbed right away. It makes it a slow death, as well as burning. At least we got it corrected as soon as possible.
I should have talked to the vet about their procedures before I ever took Maddie back in. I just didn't realize I needed to. My other vets used an IV catheter taped to the leg and injected through that, which is more stable. They also injected a sedative, first, so I was expecting her to do it that way, too. I thought that was standard practice these days, at least around here. It was my mistake. I was surprised when I saw they weren't using an IV cath, to inject through. They just injected straight into the leg. They didn't inject a sedative first, either. She was sticking her just as I realized what she had and I didn't have time to react. I was focusing on my dog.
I thought about not posting this, so people on the board facing this wouldn't be anxious about something going wrong. On the other hand, maybe this is something people should know about and be able to talk to their vets about.