Hotel Tango, it didn't originate from K4/5 but most likely from the sea-side apron of the terminal 1, i.e @ K8/9 limit.
So taxi via K, then D, then M to 25R. Only D & M parts highlighted on the diagram, but you can see the K part at the beginning of the video.
I guess that makes sense. What doesn't (to me) make any sense is a taxy route which effectively crosses an active runway no less than 3 times when the route K, J, E and M would only necessitate 1 crossing. Seems a strange procedure, albeit without knowing all the facts of course.