In just a few lines, the speaker argues that joy and touch keep us alive. A mouth can "die / for want of mirth," meaning it withers without laughter, or perhaps without speech and kisses. Skin suffers in a similar way when there is no mouth to "read" it, a nice image that treats touch as a kind of language, with lips decoding the body like a page.
The last stanza brings the two together in "a merry lay." A "lay" is a short song or poem, but the word also carries a cheeky second meaning of lovemaking, and the title's nod to John Donne suggests the poet is playing with that double sense. Judging by the title and tone, Donne's name seems to stand for witty, physical, bookish love poetry. The mood is light, teasing and warm. Beneath the humor sits a serious idea: without pleasure and closeness, something in us goes quiet, and being together brings it back to life.