Covering a caged bird I have found to be very dependent on the bird and the environment in which they are in. We used to have cockatiels in the living room, and covering those up late evening worked really well (they were a pair though). however, we then got a Ringneck, who if you covered her cage up, would do everything she could to see out, and spent most of her time at the bottom peeping through the small gap (which we always left to avoid complete darkness within) and wouldn't settle with it.
Since we have had birds in the conservatory (we don't spend our evening in there) I havn't bothered with the covers as they tend to spend their night roosting on the play gym above their cage, nor do I heat the place for the birds benefit. They are far more adapt at climate control than we are, so as long as they are not in any drafts then they will be fine, just as long as her feathers are in good condition (spray her regularly) and they are fed a good diet.
So all in all, I've not been much help to your question, try her out with a cover, and once a routine established then things should be ok. But again, as my birds are out all night, they are free to start playing when they want in a morning rather than when I get up, so they respond more to their natural habits. All the birds in the aviary lighting on a timer during the winter months for the same reason.
It's good to know she has settled with you though, and you with her.
If God had intended man to fly, He would have given us wings. Instead He gave us Parrots.