Yes I could build a bow drill kit with just a knife and what I could gather if I had to, the knife and a considerable amount of time, time and a boot string. If I had to make cordage, I'd add quite a bit more time to it. I learned all of the skills necessary in years of taking various primitive and survival skills courses.

The kit is what takes all of the time. A hand drill kit would be far easier to put together, but as I mentioned I'm not consistently able to make a fire that way. If I were going this route, I'd try a fireplow first. I have been successful with a fireplow, but it has drawbacks as well, the main one being the work you have to do to get a coal. Turning a spindle into a piece of wood is less tiresome than pushing one piece of wood back and forth into another. I can work up a big sweat on a fire plow as well.

Building a bow drill fire kit from scratch is time consuming.

First you have to find the right kind of wood for the spindle and hearth (board), and that wood should preferably be at least dry. This takes a little bit of knowledge of trees and wood. It's not like you can pick up any piece of wood to make a successful kit out of it. There are some sites on the web that show people getting coals out of all types of wood that I've never been able to use successfully.

There are a couple of things you could do here that might speed everything up if you're lucky. If you can find wood of the correct thickness, straightness and length so that you only have to strip bark for a decent spindle it will save time. If you can't find another larger piece to split for a hearth, but you have a second boot string, you can tie two pieces of the spindle branch together to make a hearth. Then if you're lucky enough to find a rock with a nice divot, a bottlecap or another piece of wood that fits well in your hand you don't have to cut a piece to make a handhold. If you use a wood handhold you might have to find something suitable to grease the piece with so it won't "burn" instead of the hearth.

Sorry about droning on there. I got a little bit off track.

The simple answer is that I could build a bow drill fire kit with just a knife. I could build one if I found a good rock to flint knap and one to use as a percussion tool to make a cutting tool. The problem is the time it would take to get everything ready. I'd prefer taking along matches or a fire steel.