A more or less literal translation would be something like "please accept my kindest regards". But I wouldn´t say that in English.

Normal ways of ending letters: Kind Regards, Yours sincerely (if no name for the addressee), Yours faithfully...

I agree with Hazel that salutations and greetings in letters are formulaic. Italian formulas translate very well into French and viceversa, but English and Italian are somewhat impermeable to these kind of high-faluting formalities. Also, as formal as Italian can be, what we have here sounds antiquated (si degni!)
This said, English has different degrees of formality, albeit much reduced.

At the site indicated below you will find the protocol used to begin and close letters for Canadian dignitaries.
I am quoting just the first paragraph for the formulas used for the Governor General, the highest ranking person in the land:
Address:
His/Her Excellency The Governor General, Government House, Ottawa.
His/Her Excellency the Rt. Hon. (name), P.C., C.C., Governor General of Canada,Government House, Ottawa
Salutation and Closing:
(Sir/Madam) I have the honour to be, Sir/Madam, Your Excellency's obedient servant,
or, (Dear Governor General) Believe me, Your Excellency, Yours sincerely,
or, (My dear Governor General) With kind regards, Yours very sincerely,