Why does my fan increase speed when the light kit is switched on?
The light and the fan are on different wall switches. If these are on different legs (180 deg out of phase), then the net current in a common neutral in the fan will be the difference in the currents in the two hots.

IIRC when the light kit is installed on the fan there is a wire nut splice of the neutral of the light to a common neutral (whites), and of course there is a splice to the hot of the light (blues). 

Suppose the fan and light have a common neutral in the fan and there is significant resistance in the neutral wire or in the splices where the light and fan neutrals join the common neutral in the light fixture or where the common neutral in the fan shroud connects to the neutral of the house wiring. When both the light and the fan are on the voltage at the fan motor and the light sockets is higher than when only one is on.

Easiest way to test this explanation is to remove all the bulbs but one from the light fixture. With the light off turn on the fan and let it come to constant speed, then turn on the light. The fan should speed up much less when all the bulbs were in their sockets.

If you can reach both of the wall switches with voltmeter leads, test the voltage across the two.hots. Zero volts indicates that the two hots are on the same leg and this explanation is wrong. 240 V means they are on different legs and this explanation is possibly true.

The remedy would be to redo the splices where the neutral come together. If the neutral wire itself is at fault, then the wire is damaged or you are using higher power bulbs (high power incandescents) than the fixture is rated for.