How to increase light from a vanity cabinet?
Fluorescent

Cram more tubes in there. Don't use T12 tubes, they are too bulky.  You can get four T8 tubes in that space if you really cram them tight.  Better still, use T5 tubes, which are almost half the size of T8.  All are equally bright to the old T12. 

48" (45-something for T5) is a good length as this will be the most common tube and ballast.  

You need ballasts matched to the length and T-code (which is the diameter in eighths of an inch).

LED

Using low voltage LED strips, you can out an essentially unlimited amount of light in there - just a case of how much 12/24V power you want to supply.  The low voltage LED strips are cheap Chinese but they are unlikely to give any trouble.  Use a UL listed power supply such as a GE or Meanwell.