Do LEDs require more power to function as they age?
The problem you are seeing is related to the construction of LED backlight circuits which require lots of LEDs. Often on poorly designed or "cheapo" designs there are LEDs or series strings of LEDs that are connected in parallel. See the typical diagram below for reference:



The problem comes into play here when the forward voltage drops of each string of LEDs are not the same. The string with the lowest forward voltage takes most of the current and is thus the brightest. In the worst case one string has a forward drop so much higher that it will not even light.

In your case what you are most likely seeing is that as the monitor has aged the forward voltage drop of some strings has either dropped or increased. Your having to increase the brightness level has to do with increasing the net drive to the parallel strings so that even those with the higher voltage drop are still able to light enough so you can see them. 

A much better design for a backlight LED array will use a separate current source or sink for each of the series string. The schematic below shows an example. This of course adds cost and a monitor manufacturer in a competitive market may choose the simpler circuit at the loss of overall quality.



This forward voltage drop phenomenon can be seen in the cheapest LED flashlights where they try to use many extremely cheap LEDs in place of a much brighter single LED. An example flashlight, as seen below, will show some LEDs start to flicker or simply stop lighting at all as the battery voltage reduces as it discharges. When LED flashlights first appeared most were built this way but as the technology has improved most devices you find today will be using the single LED.