Argh this is very difficult, isn't it?
Just to give you a parallel.
We had a situation in my department with an academic member of staff covering up a serious issue that has stopped them from fulfilling their duties properly. We tried to help this person, and have repeatedly over the years stepped in & covered for them - to get the necessary work done. It's become clear that by trying to be human & compassionate, we've all been participating in the pretense that this person is doing their job.
It came to a head at Christmas, and it was clear that not only were they not capable of doing their job, but that the stress & extra work on other staff members was having a serious cost.
So we now have a policy of doing everything by the book. We are trying to stop dealing with it ourselves, and turning the whole situation over to the University service departments (such as HR for your situation, I'd think) whose job it is to deal with it. Our colleague may not continue to work, but the situation was becoming untenable for everyone, not just them.
It's tough but I think that you may have to sort out how and where you can follow up a proper process.
I am a great believer in students being off limits. I don't buy the "But they're adults" line. Technically, yes, but the power between students and older, male staff is so unbalanced - I just despise male colleagues who take up with students: can't they cope with women(or men) their own age & status.
As HoD, I've been at the cutting edge of having to manage a situation of a relationships between male staff and much younger female students. Twice. I have nothing good to say about my two male colleagues who did this. Meeting with the female students concerned, to let them know what steps we were taking to ensure that there was no perception or actual favour and that they could be sure they were getting their results without favour - ugh, I do not want to ever have to do that again.
And I also saw the fall out with other female students. They are often fragile, and seek our approval - one young woman said to me "Why did he pick her and not me?" That to me is one of the usually unacknowledged costs of men treating female undergraduates as their sexual hunting ground.
But you have to find out what the procedure is and follow it to the letter. Require that HR, the Student Union, whoever might be an official "stakeholder" in the transparency & equity of teaching do their work. So often, we academics think we can sort it out.
I know of a senior woman academic who will attack publicly and in a full-frontal way when she comes across things of this sort, and that approach just doesn't work in my disciplinary/institutional context: this woman has lost her credibility and gets treated like an unexploded bomb
Gosh, this could be me - although I am treated with respect, when I speak out, not an unexploded bomb. I argue my case, robustly, but it is not personal attack. You don't get anywhere with that. But I think I am known for calling a spade a spade. I see it as a feminist ethic to put my views openly where my politics are.
"Feminesse oblige" as one of my mentors calls it.
On the other hand, the female gossip network is a very effective weapon. Don't think academic men don't have similar networks - they are generally far less honourable than women are; far less honest usually.
So perhaps think about how you can mobilise a gossip network without behaving badly, and without 'convicting' someone on hearsay?
Do you have a network of senior female and right-thinking male colleagues? Can you seek out a bit of advice? As someone of whom advice is often asked, to be honest, it's very flattering that people respect one's opinion.