@r042wal It seems you are ’new’ to headphones / headphone systems. I encourage you to keep the first steps simple.
Figure out which headphone(s) you like / enjoy / appreciate. The easiest way to do this is to start with a mainstream option. The advantage is the price point, ease of trial and return, up to date in terms of wireless specs/codecs, and a quick re-sale via a local site like craigslist. You can usually listen / audition at your local best buy or equivalent store. Most of these options will come with ANC, but it can be turned off. ANC does come handy and you may wish you had the option when you realize so. They are also fully portable and will not restrict you to your audio room.
If you go wired, it moves from a bit more complicated to much more complicated. You are already trying to make the step of using aspects of your current system...which can be fraught with issues and a domino effect (typically not in a good way and in dollars spent) for someone just starting out.
There are mainstream options within your budget that are both wired and wireless.
The first sentence of your opening post points to "needing your wife to enjoy her sleep." Open back headphones are typically superior sonically but they do bleed sound.
There are tradeoffs all around. And it is not straightforward.
I’ll reiterate: since you are new to this start basic. Learn. Move on with a greater understanding and to better and wider options.
All of the major players in the mainstream wireless space have decent offerings within your budget, and with budget to spare.
If you want to dig deeper, go over to headfi.org