After the full Quran was revealed, how many years did the prophet live after that?
If you mean when the Quran was completely revealed and settled by the scribes of the Prophet () and memorizers among the sahaba with the content we know today. Then the answer is simple it was fully revealed before the death of Muhammad (). Source see for example this hadith of abu Hurrairah and this of 'Aishah.
A logical approach based on the narrations from above
As Muhammad has died in rabi' al-Awal of the year 11 a.H. one could say the last revelation must have been before Ramadan of the year 10 a.H. so at most there have been one year between the last revelation and the death of the Prophet ().
Narrations quoting that some abrogated verses have still been known by some sahaba or not yet destroyed at the time of death of Muhammad () would support the possibility of at least 7 months before his death!
What the scholars say based on some narrations
Note that scholars have different opinions about the last verse(s) revealed and ibn Abi Hatim ابن أبي حاتم in his tafsir #2944 quoted that the last verse is 2:281 (Imam at-Tabarni has quoted it in his al-Kabeer here and here on the authority of ibn 'Abbas and is also quoted in Sunan an-Nasa'i al-Kubra #10992) and that it was revealed according to ibn Jubayr nine days before the Prophets death, al-Qurtobi in his tafsir attributed this statement to ibn Jurayj ابن جريج while saying that ibn Jubayr and Moqatil said it was 7 days before the Prophets death and 3 days were also quoted by some. Imam at-Tabari in his tafsir just said this is one of the verses which have been said to be the last one revealed and added several narrations reporting it and confirming also some of the statements made by al-Qurtobi. He even quoted the statement of ibn Jurayj which sounds to me as if he had some doubts as he said: "They say that ..."
I've read on the site of shi'a scholar a quote saying that the last revelation was in dhil hijjah.
Criticism and conclusion
However these statements would strengthen the position of those ahadith and narrations which say that some abrogated verses were still spread between the sahaba. They leave an open issue or contradiction: How should we understand that the Prophet () has recited the (full) Quran the last time in Ramadan if the last revelation was a few days before his death which was in Rabi' al-Awal? At the end the only thing we can say is that Quran was revealed as long as the Prophet() was alive and the revelation ended in his last year of life.
Maybe helpful a fatwa discussing whether Quran was revealed at once or gradually.