The UK has a procedure for people who have entered without an intention to marry and subsequently form an intention to marry while they are in the UK. That procedure requires notifying the Home Office, and, as noted in MJeffryes's answer, the Home Office may extend the notice period to 70 days. Then again, they might not.
You can read about the Home Office investigation in the guidance (pdf). The process does not leave room for arguments such as "you can return home so you should follow the normal procedure."
Your new wife will not be able to remain in the UK on the basis of your marriage, because it is not possible to switch from visitor status to family-based residence. But in contrast to the other answer, I would point out that getting the marriage taken care of now, while she is in the UK, will make it easier to get a spouse visa when she wants to return, because you will no longer face the need to convince the Home Office that your relationship is genuine.