If there's anything I'd like to shoot with my bow more than a coyote, it would be my neighbor's weiner dog.

If I were in your situation, I'd wait until the coyote ate the other dachsund, too, THEN I'd find some way to take down the coyote. After that, I'd buy the little neighbor girl a German Shepherd. (Or a Bulldog, or a Great Pyrennes, or a Mastiff, or a Bluetick, etc.)

Seriously, though, if the remaining weiner dog ISN'T a worthless yapping demon like my neighbor's, it may be a good idea to get a big dog to protect it. That coyote will be back once he gets hungry enough again. It worked for him once and he'll remember that.