I guess that the problem you are considering is quite hard.

First, let me point out that the similar problem where one tries to minimize the *average* (instead of maximal) distance to a connected curve of given length has been studied. It is called the irrigation problem, see for example Mosconi and Tilli's paper "$\Gamma$-convergence for the irrigation problem", *J. Convex Anal.* **12** (2005), no. 1, p. 145-158.

Note that one can twist the problem in several ways: insisting to have a loop is different from considering any curve (possibly branching: in Mosconi and Tilli a curve is simply a Hausdorff one-dimensional compact set, and its length is its one-dimensional Hausdorff measure).
In the case of a curve, I would guess that the optimizing shapes are asymptotically the same for the average distance and the maximal distance, that is to say the $1$-skeleton of an hexagonal tilling.

In both cases, one can give at least an estimation of $r$.

First, one can easily get an upper bound by constructing an explicit curve; for example, a loop that is mostly made of vertical lines at distance $2r$ one to each other. It as length of the order of $1/r$, so that you get the estimation $r\leqslant a/l$ for some constant $a$.

One also gets a lower bound by the following argument: consider a square inside your disk of roughly the same size, and divide it into subsquares of side length $3r$. Then a curve that achieve the bound $r$ must go far inside each of these cubes, and have length at least $r$ times the number of edges of a spanning tree of the incidence graph of the cube subdivision. There are roughly $r^{-2}$ vertices in this graph, thus roughly $r^{-2}$ edges in a spanning tree. It follows that you get the inequality $r\geqslant b/l$ for some constant $b$.

The problem is to get precise values for $a$ and $b$; the above strategy can give explicit values, but even with a little bit of optimization it should be difficult to have $a$ close to $b$, though $r(l)$ should be equivalent to some $c/l$.