Andrew, when I see a rhyming poem translated without rhyme, it always bothers me. Here Heine has just two lines per quatrain that rhyme, so it shouldn't be impossible to replicate if you are willing to be slightly looser in your meter or your translation. You may say that you have consonant slant rhyme in the first quatrain and assonant in the second, but they are both almost imperceptible, so if you can bring them closer to true rhyme, I think they would give that little sigh of satisfaction that is the counterweight to the sadness/frustration in the poem. Note the wide separation of the rhymes. This downplays the chiming effect without canceling it entirely.