a plant dying wont cause your nitrates to raise, it causes phosphates to raise.
As far as your nitrites keep spiking, one of 3 things is happening.
#1 Your tank hasn't reached an equilibrium of balance between the bio-load and the bacteria established to convert toxins to nitrate. I.E. your cycle isn't complete.
-Keep an eye on your ammonia and nitrite levels, do a pwc when they get in the "unsafe" zone
#2 You are cleaning too much. If you rinse off filter media, ornaments, and fake plants in tap water, and then do a gravel vac, you are probably removing a bunch of your good bacteria causing mini-cycles.
-Stop doing that
#3 Something is decaying in the tank. (Not plants) A few dead snails or fish, or overfeeding that is causing for additional ammonia above and beyond your bio-load that your bacteria cant break down causing mini-cycles.
-Remove the ammonia causing source
An established tank should not show any nitrites, and even small adjustments in bio-load or cleaning doesn't cause enough to show results in my expereience. Hope this helps