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The problem is that the 25th Amendment is for physical infirmity, not mental. Per Section 4, all Demented Joe has to do after the 25th is invoked is send two letters to resume being POTUS. It then takes a 2/3rds vote in both the House and Senate to get rid of him. (Impeachment is only a simple majority in the House and 2/3rds of the Senate)

Here is the Amendment in full:
Amendment XXV
Section 1.
In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.

Section 2.
Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.

Section 3.
Whenever the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that he is unable to discharge the powers and duties of his office, and until he transmits to them a written declaration to the contrary, such powers and duties shall be discharged by the Vice President as Acting President.

Section 4.
Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.

Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.

A survey of scholarship on the amendment found no specific threshold – medical or otherwise – for the "inability" contemplated in Section 4. The framers specifically rejected any definition of the term, prioritizing flexibility. Those implementing Section 4 should focus on whether – in an objective sense taking all of the circumstances into account – the President is "unable to discharge the powers and duties" of the office. The amendment does not require that any particular type or amount of evidence be submitted to determine that the President is unable to perform his duties. While the framers did imagine that medical evidence would be helpful to the determination of whether the President is unable, neither medical expertise nor diagnosis is required for a determination of inability ... To be sure, foremost in [the minds of the framers] was a physical or mental impairment. But the text of Section 4 sets forth a flexible standard intentionally designed to apply to a wide variety of unforeseen emergencies.[4]: 7,20 https://en.wikipedia.org/wiki/Twenty-fifth_Amendment_to_the_United_States_Constitution