Possible reasons for HKS not going in ambulance:
1) Only one person allowed to ride along. Moe may have insisted he go, he seemed very protective of ANS.
2) HKS may have known (or been convinced) that ANS was not revivable and his priority would be to stay at the room and present a suicide history or something to the police in order to attempt to influence the investigation.
3) HKS may have wanted to attempt to "control" any evidence in the room.
Probably other reasons also, just speculating. I see more reasons why HKS would prefer to stay in the hotel room than go in the ambulance. Cold-hearted, yes. That's HKS.
If he was the "purported" husband, he would have been in the ambulance. He told the EMT's on the scene he was her "husband", and many of the intitial news reports were calling him the husband, until enough made enough noise to stop it.
JMO, but if he had been truly concerned about her at that time, he would have been at the hospital/ER. They need to know a patients history, meds, etc. and he was probably the only one who knew that..
I vote for #3