In practice the only way you can settle the financial issues arising from the marriage once for all is by obtaining a court order within the context of a divorce. This can be done by agreement but no such agreement (whether agreed or not) can be approved by a court unless and until there is decree nisi in divorce. Therefore the divorce comes first or, at least, a divorce has to be progressed to at least decree nisi before you can formally and finally settle the financial issues once for all.
Under NO circumstances should you simply transfer your interest in the property to your wife except within the context of a court order obtained in divorce proceedings. If you ignore this advice you risk two consequences which are very foreseeable:-
1. Your wife might not consent to a divorce based on two years' separation and there would be nothing you could do about it.
2. Even worse, if and when there is eventually a divorce your wife might in those proceedings (and very likely would) ask for more. After all, she would already have the house. She would have nothing to lose by asking for more (and perhaps getting it because who knows what the circumstances would be at that time).
It must be done in this order otherwise you run a big risk.
As to continuing to pay half the mortgage, is this wise? For a start for as long as you do this your wife has no reason to hurry and if she thinks she is going to get the house every month more that you pay benefits her and it money from your pocket you will never get back. There is another point. She is living in and has the sole benefit of a property jointly owned by you. Why should she not be paying you rent for her use of your share of the property? The mortgage can be considered as the equivalent of rent. If she has the benefit of living in the property and you do not then why should she not pay the mortgage?
This is, of course, an argument from principle. There may be other factors which mean that you should be paying your wife maintenance (from which she can pay the mortgage). For instance, if she has a low income, you have a high income and she has children dependent on her. You do not mention any of this type of factor but where such factors exist they can have a significant effect.