You have a solid alibi and no motive for killing Mr. White, so there’s no reason for the Inspector to accuse you of the murder.  The real question is how much you should tell him about what you know.  If you fill him in on your association with Mrs. White, there’s no guarantee that he’ll keep it secret.  Besides, the fact that you were there when the man died in your brothel could make the Inspector think you were an accomplice in both murders.

On the other side of the coin, if Mrs. White was the killer, and recognizes you, you could be in danger of  being killed by her yourself.   It’s better to be exposed than dead.

Say nothing for now, and see if there’s enough evidence to hang Mrs. White without your help.  Professor Plum could be useful in finding this out.  Try to get him to tell you more about the investigation or the autopsy.  He’s a quiet sort, so you may need to ask more than once.