A lithium polymer battery can have "Cells" but at these compact sizes they are a single cell. However battery cells vary in capacity, so counting cells is not a good unit of measurement, instead you rate batteries by milliAmp-hours. or mAh. Which is the amount of amps the battery can put out over an hour of time. A 3rd cen iPod touch has a 750 mAh battery. Since the iPod doesn't pull 750mA to run, it can run for several hours on said battery. The actual amount of battery life depends the following factors: (1) What app you are running, and how much CPU that app takes. (2) How much that app accesses memory or other parts of the iPod such as the accelerometer. (3) Whether or not you have Wifi or bluetooth on, and if your app is accessing those. (4) How many apps you have running that use push notifications. (5) The brightness of your screen (6) The volume of your sound (7) The age of the battery, temperature in the room, and how well the battery was taken care of. (8) And more... As Andrew said, replacing...
The most likely cause is dying capacitors. Capacitors hold a charge of electricity over time, and are used in the power supply of the TV. Some types can fail over time due to humidity, or heat. Working on a CRT can be very dangerous due to the high voltages that can be stored inside even after the power is removed. I would try to find a local electronics repaid shop, most of them made their start working on CRT TVs and they would have the expertise to repair them. If you do want to do it yourself you have to first leave the TV unplugged for a few hours. Then take it apart, being careful not to touch any of the circuits. Then look for any blown or damaged capacitors, they would look bloated on the top of the can. Write down the specifications of each of these capacitors, find the equivalent replacements from an electronics store, and remove and re-solder them. But NONE of the above is easy, power suppliers need powerful soldering irons, and there is no guarantee the failure is the capacitors alone.
When my camera had this problem I looked for a broken model on eBay. I found a model with a broken LCD but other wise powered on. So I opened up both, removed the LCD from mine and put it into the eBay camera, the camera has worked for years after that.
Many Logitech wireless products use encrypted data transfer. However have you tried to see if Logitech sells a spare receiver? If not ask their tech-support. The receiver on my Logitech laptop mouse broke, it was 2-years old. I e-mailed their tech support and they shipped me a new one. Or you could learn to solder and try to fix the USB connector that broke on your existing receiver, all it should involve is cutting a USB cable in two, and soldering four wires. You can get a pinout of the USB cable and connector here: http://pinouts.ru/Slots/USB_pinout.shtml