You're going to have to treat it like a new cutting, more than likely it will root. I have taken cuttings and then left them in a plastic bag for 2 weeks because I forgot about them. The survivors of the mistreatment usually have a much above average rooting success.

Since the cutting seems to like the back porch maybe root it there. I would fill the bucket with a good coarse potting soil, maybe one with a lot of perlite. Apply rooting hormone to the bottom half of the cutting if you have it . Set it in bright shade, no direct hot sun. Keep the soil damp but not soaking. I know "old timers" that successfully root long cuttings in buckets full of sand, in the shade, set under a dripping faucet, this is what I am suggesting that you emulate.

Method 2 would be to find someone who is successfully rooting roses and has a mist system and beg them to root if for you.

Hopefully you have photos of the bush and flowers, so if rooting is not successful maybe we can ID the plant and you can get another one.

I wish you success.