Can you give mini dachshund aspirin?

I'm not a vet, but this sounds like a very bad idea to me. If there's any internal bleeding, the aspirin could make it much worse and possibly kill the poor dear.

My advice would be to monitor for any signs of discolouration, swelling, or limping. Skin discolouration or any swelling means you need to get to a vet right away. Limping that lasts more than an hour or so would also indicate medical attention is needed. Other than that you should be fine. Watch for any loss of appetite though. These animals can be real troopers, hiding even severe pain so as not to look weak. An appetite loss could be your only sign of severe pain due to to a non-obvious injury requiring medical attention.