Justin,
Yes - I mean Lime tree. We call them Basswood over here. The wood is extremely light and easy to get a coal with. The only consideration is that it has the tendency to "fire glaze" and get a slick shiny film while drilling and you have to break through this on occasion when it happens. Other than that, it is great wood and takes a hand drill easily.
Teasel/Willow - Could be several things. Your Teasel drill may be too hard for your Willow if the Willow is a very soft type. Not to say the combination won't work. You just have to spin faster to get the temperature up before it has a chance to work its way through the fireboard.
Probably however, it is your technique which needs modification. If you are getting only a bit of smoke and some coal. It is critical to have the right combination of speed and pressure. The pressure grinds off the powder you need for the coal while the speed is what will get the temperature up to 800 degrees farenheit (Don't know the celsius equiv).
Start out slowly until the drill and the board mate nicely and then begin to slowly put more pressure on the drill to grind off more powder. As you get the drill to smoothly work into the board, carefully tilt the top of the drill back to you a bit. This makes the tip at the bottom rotate towards the outer part of the notch and will dramatically force out a large amount of powder. Once this happens, speed up a good deal in order to get the heat up and you will see two things happen - A large amount of powder pushing out and it will begin to turn dark brown and then black along with a LOT of smoke. Get to this point and you have it made.
Earlier I had posted the tip about using a piece of Pine resin to make the drill sticky. I can't emphasize this enough as it can make all the difference in the world.
Also, if you can find some Tinder Fungus which you have there, take a piece of the dried fungus and grind to a powder. Place a pinch of the powder under the notch and as soon as any real heat touches it, pow, it will catch and begin to smoulder.
See if you can get some dried Cattail stalks and a piece of Lime (Basswood) and I think you will be happy. Teasel and Willow will be ok too. Another good wood to use is Elderberry for the drill. It is a bit hard and will drill through pretty fast and is slick and needs Pine resin but works great and has a nice pithy center which sometimes catches a coal.
BTW - Common Mullein stalks are terrific and easy to use.
Hopes this helps!