Can you do a skylight? That would be your best bet for natural daylight.

Nope, we're in the downstairs flat. Some light comes in thru a window but we need to know what the best light is for over our medicine cabinet and vanity.

thank you but i don't want my skin to look nice - i want my image in the bathroom mirror to reflect how i actually look in the daytime light - i want to see myself the way others see me. i want to apply makeup to my face and know that i won't look like a clown. i'm not looking to flatter myself . i'd like to hear from someone who is a lighting expert - what kinds of artificial light source will give me what i'm after?
if i were able to tap natural light i wouldn't be asking this question.

Here is an article with 7 natural lighting techniques for a master bathroom - hope this gives you some ideas http://blog.innovatebuildingsolutions.com/2014/11/22/7-techniques-brighten-master-bathroom/

A link to the actual fixture and length is necessary. You will also need to do a little research. It appears that a 2' fixture puts out 900 lumens as an LED, and about 2000 lumens as a T5 fluorescent. I generally recommend around 2400 lumens at the mirror. Your overhead recessed may help some. Usually, the best light for make up would be one of those fixtures mounted vertically on each side of the mirror, with a non-colored lens. In that case, the LEDs would put out enough light in maybe a 30" version. A light color temperature of 3000 Kelvin is usually OK, BUT you are asking for something closer to 5000K. You can get 5000K bulbs in both fluorescent and LED, but I am not sure what the availability of LEDs for that fixture is. I think the fluorescent might be easier to find. The color rendering index (CRI) is also something you will want to look at for this application. An LED may have a CRI of around 80. Some fluorescence lamps have an increased CRI of around 90 due to the use of tri-phosphors (rare earth phosphors).
If this was the light you would need to use for quick trips into a room, I would suggest looking for the best LED you could find. This is because the LED turns on quicker, and a fluorescent usually needs to warm up for a few seconds, and actually wastes energy for the first 15 minutes compared to a standard incandescent bulb. If the overhead light would be used for quick trips, I would go with the T5 fluorescent, mostly because I think the right bulbs will be easier to find. I think the light will also be more even with a fluorescent unless the lenses are fairly opaque, which kind of negates the color and CRI of the bulb itself..

Fred S, i respect your input (like i always do), but can you cut thru the techin-cal-a-tees and just give me the source of the light that i need ? Thanks. BTW, Happy Thanksgiving :)

I wish it could be simpler, but that is out of my control.

Seriously ? ....:))

I always advise getting lighting for bathroom mirrors, if it's vertical (about 30"x 40") to have sconces on each side in addition to whatever light there is in the bathroom. When purchasing bulbs, typically the coolness or warmth (this pertains to color) is indicated. You should choose those with 'daylight' indicated. The store clerk should be able to get you the 'daylight' kind.