To create a 'pack smell' might speed up a possible behavior change towards acceptance: stroke the face of the male with a tissue, then let the female smell it while you feed her something nice and then stroke her face with this tissue too, bring it back to him - feeding him and stroking her smell on his face at the same time (connecting the smell of the other cat with nice experiences). Stroke the parts they use to mark something
As male and female cats have quite different ways of play, it is always risky to bring them together for a longer time - and 'playing' mostly is nothing but chasing which will stress both.
As she lives (and gets to know as her territory) now a room he likes too, she will probably not be too welcome there if you open the door. Would try to get her to know other rooms too (that enables her to feels safe to go there quite quietly later, to avoid confrontations. It is important to give her the chance to avoid him, if she wants to, when both are free to share his territory). While she explores, bring him to 'her' room and close him in there for a change (that helps with the 'sameness smell' too, but bring his own 'bathroom' with him and do it only for a few minutes first).
If they seem calmer/less interested at the door and you want to have a try of letting them meet for a short time (start with minutes) during some days, choose 'cuddle time' rather than 'play time', start with the one not living in the meeting room (sometimes in her room, but better start/do it more often in his) in a cage (as you already did - well done), feed both of them some tasty treats during that time and stroke them both, then bring the 'visitor' immediately back to his/her own territory, close the door, free him/her and ignore both for some time (so that they realise: nice things happen in connection with the nearness of the other).
If all goes well (when they seem eager to interact in a friendly way - rubbing faces to the cage with the other, both of them) and you want to let them meet 'free' for a short time every now and then (might slowly get longer, if all stays well), watch out that each has its own toilett (not an open one preferably, as fear of attacks while doing his/her thing might challenge the user, force him/her to develope the habit to urinate rather in some better hidden places (in the closet, behind the TV...).
They also need their own sleeping places(!) and food in the usual place AND at some neutral places to be reached easily (toilets well hidden, places to sit and watch elevated).
Cats will share a territory (after some hissing discussions) but will use the different places there each in his/her own time. The 'loser' or the more diplomatic one goes there, when the other rests or is busy with something else.
If one of them starts to urinate 'displaced' - that usually is one of the first signs of social stress: then go a few steps back in the building of acceptance and take more time.
Sorry if this has gotten a bit long, but getting cats to accept another cat in their environment is a bit complicated and really doesn't smell nice if something goes wrong...