Which EPL match has had the longest injury time added at the end of 90 minutes?
Longest additional time: 12 minutes 58 seconds (Arsenal v. West Ham United, 23 January 2013)
Here is the match report for the game from the Guardian. The time added on was due to an injury to the West Ham player Potts.
The visitors' bleak evening was compounded when the substitute Danny Potts was carried off wearing a neck brace after almost 10 minutes of treatment, having lost consciousness in a sickening aerial collision with Bacary Sagna. The West Ham manager, Sam Allardyce, reported that Potts had "hit the ground like a dead weight" with the "double blow" being the manner in which he landed on his head.
A match between Watford and Norwich on 21 January 2022 had 15 minutes of additional time at the end, though this was due to a temporary floodlight failure causing the match to be disrupted, rather than an actual injury.
Play was suspended for 11 minutes owing to a floodlight failure [...] There were 15 minutes of stoppage time as a result of the floodlight failure...
Incidentally, the very next day, a match between Brentford and Wolverhampton had 19 minutes of additional time at the end of the first half, partly due to a collision between two Brentford players and partly due to a drone flying over the pitch that caused play to be halted. You asked for the longest additional time at the end of the second half, so this technically doesn't answer your question, but it's worth noting regardless.