He had a month to come up with a better story. He couldn't. Why?
He knew, when he confessed to police, that when they would soon examine her body that he was about to lead them to, that they would find the stab wounds (on bone). If even through her clothing, the stab holes as evidence how he killed her.
Therefore, MOO, he knew any excuse he could have made (as killers do sometimes, to make themselves look marginally less 'bad') like, "Oh, she fell. It was a terrible accident, but she fell & hit her head and died that way so I wasn't responsible" was out the window. There was no way this was going to come out at all as a tragic accident, in his favor, because the post-mortem wounds would not reflect that. He knew he was well past the excuse of it was all a big misunderstanding. I guess we have the supposed cause of death (sharp instrument of some kind) to 'thank' for that.
I still worry it may have been a head injury- swelling that maybe the ME can't determine now- but I just don't want to go there. Could be simultaneous COD. I really want this guy going away forever (none of this life equals 20 years but out earlier for "good behavior") & for her case to somehow make a difference to women in the world.... in a big way.