Thanks for the advice! I'm determined to figure it out, so will work on getting some grow lights 
I strongly recommend LED. Just so you know... sellers lie 
Expect the lights from cheapo/no-name manufacturers to only be about half as powerful as they claim they are (you can test with a watt meter). Expect cheap lights to be less efficient than quality lights. Don't trust manufacturer statements about how efficient or high quality their lights are - look for a umol/J (or umol/W/s) figure, and if it's not present, assume "inefficient". Ignore "lumens"; that's a meaningless measure for plants (the lumen scale is based on the sensitivity of the human eye). The more red in the spectrum, the more umol/J you'll get, and thus the more energy you'll give to your plants, although you have to have at least some blue to prevent excessive petiole elongation. Some "white" (aka, other colours) in the spectrum may or may not help with hormonal triggers, but again, they'll lower umol/J. A sign of quality for grow lights is "unusually large / heavy for its rated output" - more bulk means better heat dissipation, but also more cost to manufacture. When you get your lights, position them relatively close to your plant; a good "rule of thumb" is that the light should feel a little bit warm on your skin - but never hot. Good light manufacturers will often give specs on how much light will be received at what distances, but cheapo manufacturers usually won't. Your bananas would ideally like - between the windows and the grow lights - 20-30 mol/m²/d. Reflective material (such as white cloth) around your plant can help amplify the amount of light hitting it. Expect quality lights to cost a lot more than cheap lights, not just a little bit more. Do the math and decide what's right for you.
Best of luck!
(And don't forget to check for mites. Your misting at the very least is a deterrent, and as you know, bananas like moisture )
Do you have any recommendations for brand of lights? I'm on a tight budget so don't want to pay top dollar, but I definitely want something that will be worth the money I spend.
Do some math. What do you pay for power - $0,14/kWh maybe? An average LED fixture might last 50k hours. So 1kW (true power, not nominal) of fixture - about $200 from a no-name Chinese manufacturer, or $2000 from a top-end manufacturer - will use 50000 kWh over its lifespan, or $7000. So even a small efficiency difference matters a lot. The cheapo fixture might be ~1,0-1,4 umol/J while the good fixture might be 2,1-2,7 umol/J, depending on the details and spectrum. CFLs are 0,9-1,2 umol/J, while HPS is 1,4-1,8 umol/J. Also, in practice, the cheapo fixture might last 20k hours (or maybe just a couple thousand if it's really bad), while the good fixture might last more like 70k hours.
Don't think just about capital costs.
Note that HPS isn't terrible, and the fixtures are proportionally cheap. But they have a lot of downsides, including being a concentrated heat source (more of a fire/burn/rupture risk) and regular bulb burnouts (which happen in a kind of annoying manner, in that they fade, and then become difficult to strike, and can end up in a cycle where they keep trying over and over to strike). The light also, while "hormonally balanced", isn't as photosynthetically efficient as red. But HPS certainly works (before LEDs, it was the go-to lighting source for indoor cultivation, and still makes up the majority of the market - although LED eats away its market share more every year). One advantage to HPS (and especially fluorescent) over red/blue LED is that it's easier to inspect your plants. With red-blue, I recommend shutting off your lights regularly to examine your plants, because under red-blue leaves just look black... it's hard to see nutrient deficiencies, pest damage, necrosis, etc.
As for manufacturers, there's so many. A good place to start might be to search "site:reddit.com LED umol/J", as there's some really active groups discussing grow lights over there. It might give you a good jumping-off point for manufacturers.
As for what to use, I use a mixture of fixtures of different types (although nowadays overwhelmingly LED) and qualities. There's no simple answer. But I want to make sure that you take the long-term picture into account and don't just think about upfront costs. Determine your square meters, determine how many mol/m²/d you want to deliver to that area, pick your light type with respect to *all* factors and what sort of umol/J (aka umol/W/s) you can expect, decide how many hours you want your lights on per day, then the wattage you'll need is "W = (mol/m²/d) * 1000000umol/mol * m² / (3600s/h * (h/d)) / (umol/W/s)". So, for example, if you want to add half sun (~15 mol/m²/d) to what the plants get through the window, and do that over 1 square meter, for 12 hours per day, using a 2,2 umol/J fixture: (15 mol/m²/d) * 1000000umol/mol * 1m² / (3600s/h * (12 h/d)) / (2,2 umol/W/s) = 158W.
Note that you can get by on smaller fixtures if you run them for more hours per day. Note that long photoperiods can prevent flowering in short-day plants, and CAM plants don't like it in general, but most plants seem to do well with long photoperiods, if you're looking to cut capital costs.
