Sufficient evidence: witness (at least one) who witnessed all or part of the beating. Witnesses (2 or more) saw him dragging the body to the balcony, heard him utter "she wouldn't stop laughing at me".  Numerous witnesses saw him immediately after the murder and when placed under lock and key.
Aggravating circumstances - he beat her to death in front of at least one of the children, possibly more then one. Particularly cruel and devastating beating that last for a number of minutes, child (children) begging him to stop.
Mitigating ? Are you kidding ? Let's see......nope, NONE.
Excited utterances can be admittted into court, his "she wouldn't stop laughing at me" should be allowed in. not enough information regarding the evidence. If he did use a weapon (bunk ladder ?) and testimony confirms it and forensic tests prove it was used, and a witness saw him using it, then there is evidence. 
18 U.S. Code § 2261 - Interstate domestic violence