Firstly, don't beat yourself up over this. I know it is frustrating, but accidents happen when you keep animals. You can only try to mitigate as much as possible. Even under the best of circumstances, we sometimes lose animals. You can only do your best. It is hard to know all the forms of mischief they can get into.
Now, as far as Daschies go- what others have said is so true. They are hunters. They are also some of the sweetest, most lovable dogs I have ever had. I have one (did have two) that is just the most adorable dog you can imagine (to humans). When it comes to birds, she will kill every one she can get near.
I once I had a small rectangular bid cage, like a breeder Canary cage, with a button quail and about a half dozen finches. It was one with the slide up doors about 4" by 5" tall. The cage itself was probably like 15"x15"x20" long or so. Our little girl, who is herself about 25 pounds got her ENTIRE body in the cage and killed all the birds. I would not have believed had I not seen the aftermath with my own eyes.
We had another situation where we converted a covered carport to a grow-out run for our ducks. We surrounded it completely with 1x2 welded wire that was sunk into the ground with large tent stakes. We let the Daschies out into the yard and about fifteen minutes later we heard the ducks making a huge ruckus. The dogs, whom had *NEVER* dug a hole previously, had already made it inside with the ducks. Luckily, they didn't have time to catch any.
The point is- animals that hunt can be very sly and quick in getting their prey. You can only do your best and make sure to never leave them unattended. After the duck incident, we simply never, ever let the dogs outside unattended.