One of the classic lost Elizabethan rhymes is in the last verse of Thomas Morley's "Now is the month of maying":
Say, dainty nymphs, and speak,
Shall we play barley-break?
For Morley, "speak" and "break" rhymed, but any modern attempt to make them do so is bound to sound artificial unless the whole thing is being sung in period pronunciation. Even then it probably won't work. The problem with period pronunciation is that it absolutely must be done in a manner that sounds entirely natural and unselfconscious and this almost never happens. Usually it sounds like a send-up, which a genuinely natural accent won't (and we all get to hear plenty these days). So, personally, I wouldn't bother about it. If you feel really nerdy (which I admit I never have), get stuck into chapter 4 (p.90) of this excellent book.
The north Kent/SE London accent would make them pretty close to a rhyme. When we lived in Orpington my wife would occasionally comment on "queen" pronounced rather close to "quane".
A broad midlands accent does it as well.