Kodak Play Sport pocket video
I had the same problem. Specifically, using a microUSB cable to charge the camera worked fine for a few uses. One time I let the battery go very, very dead. When that happened my camera would not charge when I plugged it into the computer or charger. The light on the power button never lit up either. I couldn't find a way to fix it until I tried this . . .
The fix I am about to describe is repeatable - works every time (for me). Unfortunately it is not easy for everybody to accomplish.
On the bright side, if you do this once and fix the problem then you should not have to do it again . . . well, unless you drain the camera battery too far again in the future.
Items needed:
1: A 3-terminal 3.6v battery charger for an external lithium battery pack. The unit must have clearly marked positive (+), negative (-) and temperature (T) terminals that allow alligator clips to be connected. I have such a charger for one of my old digital cameras. Remember to check that it charges 3.6v battery packs and has 3 accessible terminals that are clearly labeled.
2: Your camera's battery pack, removed from the camera.
3: Three wires with alligator clips on both ends. Ideally each would be a different color. Black, red and yellow will be used in this example.
4: A piece of stiff clay or similar substance that can be shaped.
5: I don't want to sound snippy, but you will need awareness of the risks and rules when dealing with electricity. For example, do not short the terminals of the charger. Do not use a charger that does not match the voltage of your camera battery. There are more "do not's" than I can list here. If you are unsure, "do not" do this.
Instructions:
1: If at any point the project starts getting hot, immediately unplug the charger and disconnect the wires. Do not apply power to the charger in until electrical connectons have been properly made. Consult a knowledgable person if you have any doubts. You might void your warranty if something goes wrong.
2: Place the camera battery into the clay to immobilize it. Shape the clay to allow the contact/charging terminals of the battery to be exposed. Save a spare piece of clay for later (read ahead to see how big).
3: Clip one end of your black wire to the negative terminal of the battery charger.
4: It is impossible to "clip" the other end of the wire to the battery. Instead, use the spare piece of clay to hold the wire in a position where the correct terminal of the battery can be "leaned" against the wire.
5: Repeat steps 3 and 4 for the positive terminal of the charger and battery, using the red wire.
6: Repeat steps 3 and 4 for the "T" terminal of the charger and battery using the yellow wire.
7: Inspect the product carefully. Your project must be able to sit still without shorting, etc for about 1/2 hour or so.
8: Carefully plug in your charger, and avoid disturbing your wires.
9: The charge indicator should illuminate on the charger. If all is well, monitor (I mean really watch) your battery for about 1/2 hour while it charges up. Carefully feel the charger and battery for heat, especially at the beginning.
I don't recommend allowing it to charge for any longer than 1/2 hour. The point is to allow your battery to be charged just enough to fix the problem. It could be risky to attempt a full charge.
Theory behind the idea: The software running within the camera is responsible for configuring the USB port and internal circuitry to enter a charge cycle. That software only runs properly after the camera "boots up." In order for the boot process to complete successfully the battery must have enough power to sustain the unit for a few seconds. When the battery is "very drained" it will not supply enough power to the camera long enough to allow a complete boot process, therefore charging does not occur. This leaves the camera in an unusable state where it cannot be charged or turned on in any way. In my opinion this is a design flaw that could be easily overcome via software by raising the minimum threshold before the camera ceases to allow recording/playback.
It is a shame that Kodak has not come clean about this. For affected individuals, they may do well to ship out a new charged battery and a software update.