Simply because he can't. Light doesn't know L's name to write in the Death Note.
You also have to realise L's intelligence and the level of wits they are matching. L made tremendous precautions and implied to Light that he knew how the Note works. He had also made it clear that if he died in the near future, Light is to be the primary suspect for Kira.
Additionaly, the write has given us multiple Rules of Deathnote. Among them the following relate to this scenario. The 23 day rule plays into the fact that 23 days is in the near future and L dying in that period would make the police suspicious of Light.
- If you write die of disease with a specific disease's name and the person's time of death, there must be a sufficient amount of time for the disease to progress. If the set time is too tight, the victim will die of a heart attack after 6 minutes and 40 seconds after completing the Death Note.
- If you write, die of disease for the cause of death, but only write a specific time of death without the actual name of disease, the human will die from an adequate disease. But the Death Note can only operate within 23 days (in the human calendar). This is called the 23 day rule.
- If you write die of disease like before with a specific disease's name, but without a specific time, if it takes more than 24 days for the human to die the 23 day rule will not take effect and the human will die at an adequate time depending on the disease.
How to Use: XXVII/XXVIII
It may also be physically impossible for Deathnote to give a genetic disease. So L may still die from a Heart Attack. Though we have no idea about L's family since he was indeed an orphan.Huntington's Disease
Out-Universe Explanation You need to remember this is an Anime, and using such obscure tactics wouldn't make it easily understandable. The point is to understand this is indeed a fantasy thriller anime, that is meant to be an engaging match of wits and not knowledge. Light is just a high school student and not a medical doctor who is trying to find out obscure ways to kill the individual. Point I am trying to make is that the "Rules of Death Note" may be a writing device to cover such plot holes, but they may or may not be enough to cover all.