> Would you mind providing some details, please? 

Is perhaps the more 'correct' and slightly more formal way of rephrasing your question. Or the example given by @unorthodox grammar is just as good, and slightly less formal:

> If you don't mind, would you please provide some details...

Alternatively:

> Would you mind? Could you (please) provide some details....

Is a bit more conversational. (Not something you would write in a formal email). Both of the examples you gave I have seen written or heard in everyday conversation, and only now having thought about it do they sound slightly irregular.