Which expression is stronger? "My every attempt" or "every attempt of mine"?

It must be reckoned that the first form (my every attempt) is largely in favour since the 1980's, whereas from the 1860's to that date there was no marked preference for either (ngram).
Personally, I don't feel any inclination to use one rather than the other, although I use the second one naturally without thinking of the other one. No criterion of relative elegance seem to impinge on my thinking about those form, but of course I can't pretend to be in possession of all relevant facts necessary for a full appreciation.

If we look a little deeper into the question we find that the ngram finding above does not reflect a universal pattern.

book second form largely dominant
poem second form only
record neither form is used, "all my records" instead
horse only the second form
dog neither form is used, "all my dogs" instead

Those few tests show that in the end the second form is preferred nowadays, but, again, that is not an indubitable proof of superior elegance for the second form.