The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
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.
Best Answer
So, we have a lamp with the following specifications:
T8 15 Watt 1"/25.4mm dia tube with ANSI G13 medium BiPin socket having a 3000 kelvin Warm White color rating and a CRI (Color Rendering Index) of 50 that is nominally 18"/455mm long.
So, basically look for the following tubes and check the length. Narva measures from end cap to end cap while Satco uses Maximum Overall Length which includes the pins in the length.