Impeachment probably isn't 'jeopardy' for Double Jeopardy purposes
The Fifth Amendment reads, in relevant part: [N]or shall any person be subject for the same offence to be twice put in jeopardy of life or limb...
"Of life or limb" is interpreted broadly to mean any criminal charge. Which impeachment is not. If convicted of a crime, you might suffer a wide variety of possible criminal penalties, from execution all the way down to community service or a fine.
But none of those are at issue if you are convicted in an impeachment. The only two possible results of being impeached, and then convicted, are removal from office and/or being barred from holding any office in the future. Being fired, in essence. That's not a criminal penalty.
Practical concerns make it unlikely for impeachment to be repeated, but there's nothing stopping the House from deciding it didn't like the way the Senate tried the case, then impeaching the president again on the exact same charge. It would be unwise, but not illegal.