Since the are you need to tile is not that big at all you could easily just list all the tilings (which is definitely not a pretty way to solve it but if you have no idea how to start it at least gives you a starting point).
Using an odd number of $L$ shaped tile you won´t succeed so you can directly start checking the cases where you have an even number of $L$ tiles.
Using $0$ of the $L$ tiles you will be able to tile it using only the $I$ shaped ones and this can be done in only $1$ way.
Using $2$ of the $L$ shaped ones you have $4$ cases,
1) they are next to each other
2) they have $2$ of the $I$ shaped ones inbetween them
3) they have $4$ of the $I$ shaped ones inbetween them
4) they have $6$ of the $I$ shaped ones inbetween them
Observe that in each of these cases you get a factor of $2$ in the number of solutions since each of these can be flipped around a horisontal axis.
And so on...
As I said this is definitely not a nice solution, creating a recurrence relation is a lot nicer altough I am unsure if you know how to use recurrence relation to solve this. Please let me know if you do, in that case we could work out something more mathematical :)
Hope this helped.