With a bright sunny room it is hard to see why your jumping spider would need additional lighting. So lets dive right in and I’ll explain why I recommend them.
Facts: Florida gets 237 days of sunshine a year
The UK gets just 1400 hours… that’s 58.3 days
The average UV level in the UK is 6
The average UV level in Florida is 10
While you can make out the 4 pairs of eyes on a jumping spider, you might be surprised to learn that 3 of those pairs see blurred greys and are designed to detect motion. Those big cartoon eyes at the very front are much sharper, and can see a range of colours, and depth which is how they can hunt so well. However, while the human eye is 10 times more effective that any of those on your jumping spider, they can see a range of colours beyond what ours can. This means that they have adapted to hunt using a wider spectrum but at the same time those two big eyes need enough light to hunt and the side pairs need to see clearly enough to feel safe. So yes, your American spider living in the gloomy UK will need a light to make them feel at home. It is also observed that the lack of strong light in a jumping spider enclosure may cause them to think there is rain on the horizon and so they will sit in their hammock instead of coming out to hunt.
As a jumping spider does not need anything beyond a simple and cheap LED, you haven’t any reason not to get one.
When looking for the ideal light you can get something simple like a desk lamp, or LED lighting strips to stick to a shelf above. You will want to look for something that says ultra-bright white or daylight in its description as this will offer a strong light with an equal spectrum of colour. After this it really comes down to a style you like and the budget you want to spend. I have LED ultra-bright strips that are automated to turn on at 6am and off at 6pm. Your spider doesn’t need this light on 24 hours, a simple day and night schedule will work beautifully: so 12 hours on and 12 hours off. You can buy plugs that can be programmed to turn the light off and on without you needing to be around. Something to note on lighting is a lamp that has a filament bulb will get warm. This can be very drying on an enclosure and without a thermostat like a heating mat has, it won’t turn off if the enclosure gets too hot which can cause moulting issues and also death. For a simple solution always buy LED. Alternatively bulbs developed for use in vivarium’s and used in conjunction with a thermostat in place of a heating mat.
I personally do not have any experience with heat lamps but there are many Facebook groups out there with keepers who use them. Please do your research fully before committing to heat bulbs to ensure you can maintain a safe humidity as they can be extremely drying, something that isn't good for sub-tropical spiders.
Fun lighting, such as UV or coloured LEDs, can make a jumping spiders enclosure look great but it may not be nice for your spider. Most coloured LEDs won’t be bright enough for day use and using them over the night might actually be intrusive to your spider. As their eyesight is more complicated than ours and there haven’t been many studies into light colours and behaviours, I would be hesitant in using them as to a spider the colour spectrum you use may look like 24grs of daylight to them. Spider eyesight is fascinating and I loved this article which wasn't overly scientific. I also think that not only is additional lighting beneficial to your spider but it makes their enclosure look more appealing to us humans.