Yes, I've seen "The Cage", I just don't think that line tallies with virtually everything else we've seen about the way warp travel works even in TOS, and should probably be ignored.What I am going by, is Navigator Jose Tyler's comment that the time barrier had been broken. Where I get my twenty fold improvement is from Kirk's comment about how long freighters take to get out there. If the time barrier is broken, then it makes more sense to have done a series about the after affects of that breaking. Twenty fold improvement in speed, is a radical change. Furthermore I am looking at the specifications of the original USS Yorktown, having a maximum speed of 6,400c. I postulate that a 'time warp ' contracts time in a similar way that a space warp contracts linear distance. I am also taking into account another comment about the early Galactic Survey Cruiser Valiant, having weak impulse power. Which implies directly low acceleration, meaning that it took time to get up to speed. Hence the need for a time warp. This implies that the four days for the Enterprise NX-01 to get to the Klingon home world, is reasonable. "Acceleration " is a component to the warp field.
6400c would be absurdly fast in Star Trek terms – around warp 9.985 on the TNG scale and over ten times the original 1701's top speed of warp 8 on the TOS scale.