With a bit of trig, the shadow at noon gives you the angle of the Sun above the horizon as it crosses the meridian (the north-south line).
Comparing the two angles will give you the angle that the sun moves north and south of the equator each year.
Once you know how far north the sun comes in the summer, you can compare that to the altitude in the sky you've already calculated, and deduce your latitude.

Apologies for not doing the calc for you, but I have a sense that I may be helping someone with their homework.

Grant Hutchison