Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we have $area(S)\ge area(D)$. Assume that $area(S)< area(D)+\delta$ where $\delta>0$ is small.
Then $S$ is close to $D$ in the following sense: there is a 3-dimensional surface $F$ filling the gap between $S$ and $D$ such that $volume(F)<\varepsilon(\delta)$ where $\varepsilon(\delta)\to 0$ as $\delta\to 0$ ($n$ is fixed). "Filling the gap" means that $\partial F=S-D$.
This fact immediately follows from the compactness theorem for flat norms. But this proof is indirect and does not answer the following questions (I am especially interested in the second one):
1) Are there explicit upper bounds for $\varepsilon(\delta)$? How do they depend on $\delta$ and $n$?
2) Can $\varepsilon(\delta)$ be independent of $n$? Or, equivalently, does the above fact hold true in the Hilbert space?
In the unlikely event that 2-dimensional surfaces are somehow special, what about the same questions about $m$-dimensional surfaces, for a fixed $m$?
Remarks: "Surfaces" here are Lipschitz surfaces or rectifiable currents or whatever you prefer to see in this context. Rather than talking about the filling surface $F$, one could equivalently say that the integral flat norm of $S-D$ is less than $\varepsilon(\delta)$.