Can I use flexible LED strips to get better lighting in my kitchen?
You can definitely get flexible led strips that are bright enough to do what you want. Many of them run on 12V, which you can easily get from a small power supply, and you could just plug it into an outlet on the counter.

I would put them under the cabinets; it would be easily to hide them there. Many of the strips can be had with adhesive backing, which would just stick under the counter. 

To answer your question on the individual lights, the individual LEDs will blend together nicely.

If you look at Amazon, you'll find a lot of solutions, and they are fairly cheap.