I thought the death penalty was just for Murder 1, maybe 2 (or whatever it is called in Texas).

If all the other things are true, none relate to murder, but shows the guy does belong in prison or a mental ward.

So, the argument should be "did he shake is daughter and did that kill the child". I cannot answer that question, but without a 100% true answer I think they should at least delay it or change it to life. You know, the whole "reasonable doubt thing". Seems many experts do have doubt, so...