Should I put my dog down for killing my chihuahua.?

First of all, really not at all your fault. Second of all, no second thoughts you need to put the dog down. It is a threat on the lives of other dogs and other children to whatever home you give her to. If you do decide to give her away, whatever foolish idiot thinks its ok and takes her, needs to know in detail the vicious nature of the dogs jealous tendencies. Font risk anyone else, do the right thing, say your goodbyes and put her down.