Chasing an archer who doesn't make a mistake is pointless. The best way to catch and archer is to anticipate their attack (most of them are pretty predictable. Winter stroke when you get too close etc) and get a precise block off.
I get bored very quickly waiting behind trees. So I prefer just throwing on defensive stance and chasing them forever. Most of the time, you'll get a random block or resist that could allow you to catch the archer, or they will make a mistake. When they screw up, feint with a spear, disable limb, and put on defensive support. After catching them it's holding them down long enough that tends to be the bigger problem for me. I try hard to get disable limb off, and then make myself as hard to CC as possible. If they cannot cc you they can't build distance, and that is when an archer dies. Always keep a spear with balestra, I keep ribs breaker 1 too, just for a few more seconds of immobility.
The most annoying thing about a knight for an archer is how hard you are to kill. I've chased archers for over 45 minutes before they finally kill me, and a lot of people will give up before that happens because it's so boring, or if it's oreo he will get close to make it more interesting
If you get really low on health, it's funny to go behind a big rock and use a health potion, that usually makes people give up completely. I'm quite sad now that since the 'quivers' have been put in it doesn't cost archers 300K in arrows to kill a knight anymore. Repairs? ^^
Anyway, that is my input on Archer Vs. Knight. For what it's worth, I barely go to war anymore on my knight, and never 1 vs 1 at this point.