It is my understanding of the Regency Act 1937 that the Sovereign cannot appoint a regent:
Regency during total incapacity of the Sovereign
If the following persons or any three or more of them, that is to say, the wife or husband of the Sovereign, the Lord Chancellor, the Speaker of the House of Commons, the Lord Chief Justice of England, and the Master of the Rolls, declare in writing that they are satisfied by evidence which shall include the evidence of physicians that the Sovereign is by reason of infirmity of mind or body incapable for the time being of performing the royal functions or that they are satisfied by evidence that the Sovereign is for some definite cause not available for the performance of those functions, then, until it is declared in like manner that His Majesty has so far recovered His health as to warrant His resumption of the royal functions or has become available for the performance thereof, as the case may be, those functions shall be performed in the name and on behalf of the Sovereign by a Regent.
The Sovereign does have power to delegate royal functions to Counsellors of State:
Power to delegate royal functions to Counsellors of State
In the event of illness not amounting to such infirmity of mind or body as is mentioned in section two of this Act ... the Sovereign may, in order to prevent delay or difficulty in the despatch of public business, by Letters Patent under the Great Seal, delegate, for the period of that illness or absence, to Counsellors of State such of the royal functions as may be specified in the Letters Patent, and may in like manner revoke or vary any such delegation …
… the Counsellors of State shall be the wife or husband of the Sovereign (if the Sovereign is married), and the four persons who, excluding any persons disqualified under this section, are next in the line of succession to the Crown …
… which is what The Queen is doing.