What should I do if I attended a conference but forgot to register?
I would suggest to contact the conference organizing committee as soon as possible and ask them what can be done about it. You can explain what happened and why, and also directly offer to pay the registration fee afterwards. If you presented at the conference and also chaired a session, there should be enough evidence that you were actually there. Conference committees are human too, and will understand that mistakes happen. If you are honest, it should be possible to convince them that you did not intentionally try to sneak into the conference or avoid the fees.

Note that many conferences will only publish your contributions if you have a valid registration and also paid the fees, so it might definitely be in your own interest to fix this mistake.

As the comment by @Roland suggests, it could also be helpful to contact your administration responsible for the reimbursement and to ask for their advice on how they want the situation to be handled. Since you are worried what impression you might give them, this should rather help to have them understand that you actively try to correct your mistake.