Hash collision resistance of $\mathcal H^\prime(m) = \mathcal H(\mathcal H(m)|m)$
Your scheme is twice as slow as the original hash function.  If you were going to accept such a huge performance penalty, you'd be better off just picking a new hash function that (you trust) isn't broken.

Also, it's not clear to me how much security your construction actually adds.  I don't immediately see a proof that it is secure even if the compression function has some problems.  I'm not even sure how one would formalize this.  We'd have to assume something about the compression function, as you can't get something from nothing; but it's not clear what minimal assumptions about the compression function are sufficient to ensure that your construction is secure.