One reason bulbs can burn out quickly is if the voltage applied to them is higher than the expected voltage (120V in The USA). Wiring problems and bad transformers can cause the voltage to be out of spec. Another reason is if there is a loose connection somewhere, and the light flickers (causing unnecessary heating/cooling cycles). A third reason is if the light is in a confined space, and overheats.
In order to check the line voltage, you will need to use a voltmeter. The Kill-A-Watt is a very safe product to use to check the line voltage. Note that you should try a few different outlets, since there are generally two different phases of power (not really phases, but opposite polarities, 180 degrees apart) going into a typical house, and you need to check both of them.
Identifying a bad connection is trickier. Generally, you would see flickering lights, but to be sure you would want to use an oscilloscope or spectrum analyzer (expensive equipment). Sometimes a bad switch will cause a short flicker when the light is turned on. Replacing the light switches with new high quality switches could help.
Finally, have you noticed that the bulbs go out more quickly in certain places? Are those places more confined, causing the bulbs to heat up more? If so, you need to use lower wattage lights (such as CFL, halogen, or less bright incandescents) so that the bulb is less hot.
LED lights, halogens, and CFLs are all more efficient than incandescent lights. For the same amount of light output, they use less electricity, and generate less waste heat (keeping them cooler). In order to check if they meet your needs, compare the light output (in lumens) to that of the bulbs you are currently using. Don't go by the "equivalent" rating (such as 60W equivalent).... The amount of heat generated is nearly proportional to the wattage of the bulb (generally between 10 and 25 W). Another thing to look at is the color temperature, as measured in Kelvins. More orange lights will be around 2700 K, while more blue (daylight) bulbs will be around 5000 K.
PS: Advertised bulb lifetimes are usually assuming use of perhaps four hours per day. So, a seven year bulb may only last a bit over a year if it is left turned on continuously.
NO PUTTY!
You have a loose electrical connection and a loose mechanical connection.
Mechanical
First you need to fix the mechanical attachment mechanism. It may be a loose nut on a threaded pipe holding the individual fixture to the base. It could be some other locking mechanism, such as bolts or screws. In some fixtures, the sockets are held in by pressure fits, which are harder to repair.
In either case, you probably need to take at least the lighting unit off of the fan, and maybe the whole fan down, depending on how it is put together. Before you do anything, turn off the circuit to the fan, not just the wall switch or the remote control switch. Once the wiring is exposed, double check that all wires are not hot by using a non-contact tester.
You need to restore the original attachment method, not just a glue or putty to wedge it in place. Fan vibration will almost certainly cause a makeshift repair to break again.
Electrical
It sounds as if you have a loose wire. You need to make sure that all wires, hot neutral, and ground are all connected (in the US, these are black or red, white, and green or bare). The ground may be established by physical contacts among the metal parts, and restoring the mechanical connection may fix an intermittent ground. Sometimes a wire may be under a wire nut, but not well connected. If there is no obvious disconnected wire, remove and check wire connections under wire nuts.
Replacement
If you cannot make the physical socket stable as originally designed, you need to replace the fixture (or at least that part of it). Underwriters approval (a safety standard) is dependent on the structure as designed. If you cannot restore the mechanical structure, you probably need to replace the entire fan.
Best Answer
Modern retrofits do not use a ballast at all like this. Most have universal drivers, they can run on 100-277 VAC 50Hz or 60Hz. Here is another example. Things to think about, the color of light 5000K will be a very white closer to sunlight, 3000K is more yellow orange. With the same lumens I think the 5000K and higher wavelength lights look brighter with the same lumen count. The type of lamp is the part that gobbles up the energy. florescent are more efficient than incandescent largely because they do not generate large amounts of heat to create light. florescent lights work on a gas plasma. LED's work with a diode junction. As technology has evolved the amount of light a LED can produce per watt of energy consumed has increased many times. There are many types of Ballast's and the type of Lamp you have will determine the type. The majority of homes have 4' T8 lamps (1" in diameter tube) some are T12 1-1/2" tube in both 4' & 8'. some of the retro kits have a strip of lights screwed to the fixture some have tubes similar to the florescent. Investigate a bit more and you will find there are hundreds of options out there and most don't use the ballast.