The Apple Music Family Membership is based on the Family Sharing feature. The 'head of the family' (forgive the archaic language), which Apple calls the
Family Organiser, is billed for the family Apple Music subscription.
So to change who pays for your family subscription, you need to change who the family organiser is. Note, this will mean that your wife would then end up paying not just for everyone's Apple Music, but also all their purchases in the App Store and iTunes Store (I assume, presently,
you pay for that stuff).
If you still want to go ahead, to change who the family organiser is, you must disband your family group by going to Settings > iCloud > Family, tap your name as the organiser, then tap Stop Family Sharing. Full instructions are
here.
All the content on each family member's device will stay as it is - it just will no longer share. For example, if your wife shared a calendar with you via family sharing, it would still be on her phone, but you would no longer see it.
Then, your wife can re-create a family group and re-subscribe to Apple Music.
The downside, as
@DeltaMac said, is that your purchased content, e.g. apps and music, will be tied to your personal Apple ID, and will have to be re-purchased under your wife's Apple ID. But if you have little or no paid apps or other content, this might not be an issue.
There is almost zero chance of getting Apple to transfer purchases from your Apple ID to your wife's, unfortunately.
EDIT: A far easier, though imperfect, solution: change your billing information on your (i.e. the family organiser's) Apple ID to your wife's credit/debit card. Then she will pay for the subscription whilst everything else stays intact.