Not every distance set $l_k$ will give you a genuine $k$-simplex. But the theorem gives you a realization in $E$ of *every* set of distances $l_k$, including the ones coming from genuine $k$-simplices. There are a bunch of $T_{l_k}$'s, some of which are $k$-simplices, and they're all nonempty (when the conditions of the theorem are satisfied).

[Complete rewrite in response to comments, having realized where my previous misunderstanding was.]