As Noah says, the direct successor of Gentzen's method, cut-elimination, has been generalized up to $\Pi^1_2$-comprehension. This was shown separately by Rathjen and Arai; the full results have never been published, but fragments have appeared in various papers. Rathjen published "An ordinal analysis of parameter free $\Pi^1_2$-comprehension", which covers a fairly strong subtheory of $\Pi^1_2$-comprehension. The strongest results of Arai's I'm finding actually published only go up to $\Pi_3$ reflection, "Proof theory for theories of ordinals II: $\Pi_3$-reflection". My paper, "Ordinal analysis by transformations", is rather vague about the comparison between the system it's analyzing and usual hierarchy, but I now think the system it analyzes is around the strength of parameter free $\Pi^1_2$-comprehension.

However the consistency of analysis has been shown by other means. First, as Carl mentions, Spector's proved the consistency of analysis in "Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics". Spector's proof is considered non-constructive (it uses bar recursion).

Girard gave a constructive consistency proof of analysis: he proves strong normalization proof for System F, which is an equivalent system. This proof can be found written up quite nicely in his book, Proofs and Types.