Any color temperature you want, really. Just keep in mind that you want to avoid all sources of bluish alerting bright light at night so that you don't prevent your production of melatonin from happening. This means that you are using f.lux as a way to enable you to use your computer without being blasted by alerting light. If you're still being blasted by alerting light from other things in your home right before you go to bed (like in the last 2-3 hours of your day), then f.lux can almost be pointless.
So let's say you've decided to avoid watching T.V. at night because you can't have f.lux on your T.V. That would be a good idea. Let's say you decided to have special light fixtures that have VERY warm light bulbs in them and these are the only sources of light you use at night. That would be another great idea! Now, let's say that you don't know about f.lux yet and you go to use your computer and you get blasted by this bright blue light. Ouch. Now what? That's where f.lux comes in. It enables you to use your computer at night without being "woken up".
During the day, use whatever you want. I just leave mine at 6500K. It doesn't matter what you use, as long as you like it.
Regarding the Intel HD Graphics Control Panel settings, I don't know if the brightness is actually controlling the backlight's brightness. It should be because it's a built-in display, but it's worth seeing if you have alternative ways to adjust the brightness. If you have other ways, then compare both to see which one is clearly actually adjusting the backlight's brightness and use that adjustment only (you'll be able to tell - one will look better when it dims). You want to use the lowest brightness that you can tolerate for the best results when you're using f.lux at night to avoid blue light.
It doesn't end there, though. Loud sounds are alerting too, and so is hard work with your muscles like exercise or doing physically-difficult things.
All in all, you just want your at-night experience to be like it is outside when the sun sets. Low light with no blue light, and nice and quiet and peaceful. You want to put yourself to sleep gently, not suddenly. Most people try to go right to bed after doing a lot of things like watching TV, playing video games, listening to good music, and then doing things in the bathroom with their super bright vanity lights on. Some people also don't know how to slow down at night and move around calmly and in a relaxed way. It's like they're always in a hurry and then they wonder why they have a hard time falling asleep. lol
So another thing to know is, you kinda want to enable the body's core to cool down and you want the mind to calm down too. It's not just about light.