At first, the second condition follows from the first by averaging over all $k$-sets containing $a$. Moreover, for any $m\leqslant k$ and any $m$-set $A\subset \{1,\dots,n\}$ we may count the number $N$ of pairs $B\subset C$ where $A\subset B$, $B$ is a $k$-set and $C$ is an $\ell$-set from $\mathcal{L}$. For any fixed $B$ there exists unique such pair, thus $N=\binom{n-m}{k-m}$. On the other hand, for fixed $C$ there exists exactly $\binom{\ell-m}{k-m}$, so $A$ belongs to exactly $\binom{n-m}{k-m}/\binom{\ell-m}{k-m}$ sets $C\in \mathcal{L}$. This ratio should be integer for all $m=1,2,\dots,k-1$, that is called divisibility condition. They are enjoyed for infinitely many $n$ (say, for all $n$ which are congruent to $\ell$ modulo $lcm\{(k-m)!\binom{\ell-m}{k-m},m=1,\dots,k-1\}$.
At second, under divisibility conditions this construction (Steiner system) exists for large enough $n$, as was proved recently by P. Keevash. See for example