Hard too say if the nitirte->nitrate bacteria has formed but there must be some present.
What i would do is,do a water change try too get the nitrates down too 0ppm, then add your ammonia too get a level of 5ppm.
Let it go for 24hours, if the cycle is complete the ammonia should be zero, along with the nitrite but cause we cant test for this, you'll have too go by your nitrate readings.
After the 24 hours your ammonia is at 0ppm, your nitrates should be over 20ppm. Just as an estamite.
Let us know how you go.
Also kinda weird that you have a nitrate test kit and not a nitrite? Normally its the other way around.. :?