I have understood the commutator relationship between $L^2$ and any $L$ component namely $L_x, L_y, L_z$. Also the fact that commutator of these two components would not commute.
However I find it difficult to relate to the uncertainty principle as why it is impossible to measure precisely any non-zero $L$ of a particle.
This is the uncertainty principle for two arbitrary observables $F$ and $G$
$$\Delta F \Delta G \geqslant \left| \frac{\langle[F,G]\rangle}{2i} \right| $$
Where $ \langle \rangle$ denotes an expectation value.
In your case you have $\langle [L_x,L_y] \rangle \neq 0 $. Thus you can't measure those 2 observables simultaneously and therefore you can't know the $L$ since you can only measure one component at a time with the precision you want, and not both of them. To dermine $L$ completely you would have to know all of its components.
Since, instead, $L^2$ commutes with $L_i$, $ (i=x,y,z)$, you can always measure the absolute value of the angular momentum and its projection on an axis you choose.
