I'd consider alternative solutions over mirrors. These also cause glare in your living room area, meaning you might get blinding reflections into your seating area or on the TV.
The angle of the sun varies both during the day and over the year (how much depends on your location/latitude). For that reason, anything reflective should additionally be tracking the sun, to keep said glare on your plants (and out of your eyes). In your case most of the sunlight entering your room was additionally reflected off of nearby buildings, which would make it an even more challenging project.
I would therefore suggest two alternatives:
- Move that plant lattice close(r) to the window
You can screw together a simple wooden frame, or put it on the back of an existing bookcase or cupboard. It depends on your room layout of course if this will work, but indoor freestanding vertical gardens are definitely a thing now.
The best would be something on wheels. Because in our case the difference between high summer and the dark of winter is a factor 6 in solar irradiation at solar noon, we move some of our plants near the windows in winter, and deep in the room in summer.
You can visually establish what they need by considering if they're getting light green or dark green (a lot of plants change the amount of chlorophyl depending on available light), and if leaves start withering at a faster rate than usual (radiators underneath windows in winter for one can cause this).
- Use plants that thrive in indirect sunlight.
There are a lot of forest dwellers that don't normally see much sunlight. These could be ferns, orchids and plenty of others. Specifically for climbing plants this site has a few dozen suggestions. I'll list a couple from there that I personally like (do browse their full list though):
- Heartleaf Philodendron (Philodendron hederaceum)
- Arrowhead Plant (Syngonium podophyllum)
- Swiss Cheese Plant (Monstera adansonii)
- Teddy Bear Vine (Cyanotis kewensis)
- String of Pearls (Senecio rowleyanus)
- String of Nickels (Dischidia nummularia)
- Pothos (Epipremnum aureum)
- English Ivy (Hedera helix)
Note that depending on your location, not all of these might be available (I'm sure that all of them left to their own devices would become an invasive species).
Finally, it's not an either or situation, you can both move the lattice to a more favourable location (which would help the plants in the dark of winter) AND choose plants that grow better in indirect light conditions.