I'm so sorry for your loss of your little Dachshund. Dachshunds are one of my favorite dog breeds and I'm sure she was the love of your life.
Dachshund veins are notoriously difficult to "hit" as they are small, roll around, and don't go in a nice straight line because of their short, curvy legs. I suspect your vet was as upset as you are. Then if your dog was sick on top of it or had heart problems, the vein will be even smaller due to dehydration or poor circulation.
Was your dog struggling or making any noise during the injections? Did the vet give a sedative at all prior to the euthanasia?
The euthanasia solution we use is an overdose of barbituates--similar to what was used as a general anesthetic at one time. So if some of the solution got in, her brain would have been anesthetized and she wouldn't have felt any more pain or discomfort or really been aware of what was going on.
Then, if her heart wasn't pumping very effectively, it would have taXXXXX XXXXXer for the thick liquid to get pumped to her internal organs and for the heart to eventually stop.
If she wasn't struggling or crying and seemed fairly sedate, then I would say that her brain was anesthetized and that she wasn't suffering at that point.