Well, 'which' works, but it doesn't seem right. Remember that 'which' must be used for non-restrictive clauses -- those that add information that is not necessary. And, for these, we need a comma. For restrictive clauses, 'that' seems to be preferred by many -- although certainly not all -- references.
Although we don't have the whole sentence, I think that the distance mentioned here heads a restrictive clause. 'That' feels better, I think.
Maybe the real reason that 'the distance (that) we've come...' seems better to me is that 'the distance (that) we've come' feels like a pat phrase.
Please click on the attachment. This is a copy of a posting in the Grammar Exchange Archives at 'that' --> 'which.'
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