I would strongly encourage looking at the wiki towers design. What you should do is this, a design that worked with me with special density balsa, but would be absolutely strong enough with regular density balsa: base braced 4 times with x's, 16 cm to 4.5 cm at the top. The x's each span 4.5 cm of main legs making a base that is 16 cm tall. Then, make a chimney/top that is 55 cm tall braced with diagonals every 3.5 cm (if you are unsure with it, go with every 3 cm which should hold all 15 kg), but no lateral pieces are required. The main pieces of the skeleton are 1/8 squared balsa and for cross bracings it is 1/16th square balsa. It should look like this:
for the base, but the top portion should be braced twice instead of once, making there be 4 x's
It should look like this:
for the top, braced every 3.5 or 3 cm with no laterals:
With this design, the tower should hold all 15 kg and be about 10-11 grams, 12-15 grams compensating for excessive glue use (especially when joining the base and top, don't use too much, but use enough).
My state tower was 8.88 grams and it held all 15 kg at my school, broke early at comp. because of transport issues, but you should be fine.
EDIT: for your cutting device, I recommend that you use a smaller blade, yet still the same sharpness. About 1-2 inches in length. This way you have more control over the cuts and you can be accurate and precise.