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Who qualified him?

But I was speaking of the hypothetical case where a person was elected and took office, but then was found to not be eligible.

Under US Law and the US Constitution, there are four required steps to qualifying a president: (1) registered voters vote for Electors in the general election (2) a candidate receives a majority of the Electoral College votes; (3) a candidate is certified by the President of the Senate (the Vice President) at a joint session of Congress as the person who received the most Electoral votes. There is a process in the US Code of Laws to object to the certification of the Electoral votes of an ineligible candidate or an ineligible elector. And (4) the certified winner is sworn in on Inauguration Day.
The Constitution and US Law is mute on any other process except impeachment, trial and conviction to remove a person from the presidency once they assume the office.
An ineligible person in the office of president can be tried in the Senate for high crimes and misdemeanors and removed by the votes of 67 Senators.

You left out the small matter of first being eligible to hold the office.

There is a process in the US Code of Laws to object to the certification of the Electoral votes of an ineligible candidate or an ineligible elector. And (4) the certified winner is sworn in on Inauguration Day.

Actually I've read that section of the law, 3 USC 15 and it provides for objecting to the vote count from a state (the votes themselves are given in the states, and indirectly to the votes of the electors. It's about process and not at all about eligibility of the Candidates. The grounds for objection are very narrow, and do not include the qualification of the candidates. But the law does say that the President of the Senate (the "old" Vice President) is to call for objections, which he did not. Here is the relevant section for those not wishing to click on the link.

Upon such reading of any such certificate or paper, the President of the Senate shall call for objections, if any. Every objection shall be made in writing, and shall state clearly and concisely, and without argument, the ground thereof, and shall be signed by at least one Senator and one Member of the House of Representatives before the same shall be received. When all objections so made to any vote or paper from a State shall have been received and read, the Senate shall thereupon withdraw, and such objections shall be submitted to the Senate for its decision; and the Speaker of the House of Representatives shall, in like manner, submit such objections to the House of Representatives for its decision; and no electoral vote or votes from any State which shall have been regularly given by electors whose appointment has been lawfully certified to according to section 6 of this title from which but one return has been received shall be rejected, but the two Houses concurrently may reject the vote or votes when they agree that such vote or votes have not been so regularly given by electors whose appointment has been so certified.