I agree - do 100% change on the tank to "start over" on the nitrates. Do not do anything to your filter media - other than rinse it in used fish water or in treated fresh water...... that way you can remove any mulm and waste, but should not disturb the beneficial bacteria that populate the media.
As far as gravel..... many people go "bare bottom" - using the minimum of stones - larger ones, too - so that the fish cannot be harmed by swallowing stones. If a fish can fit it in his mouth, he will. In fact - if a fish even THINKS he can fit it in his mouth, he will try!
Having much less on the bottom of the tank means far less waste will be trapped in the gravel..... (less disease and parasites, too). You will be able to totally clean the tank. Removing excess waste from the gravel before it is transformed into nitrates will also help in your weekly nitrate levels.....
Sometimes, when the nitrate levels get so high, the nitrogen cycle will slow - so test your other parameters, too. Those BB do not necessarily like to work so efficiantly in high levels of nitrate (nitrite/ammonia, too).
I hope it is as simple as a bad test kit..... but, in case it is not, you need to have a look at how you change out your water. Lets take a look at some numbers....
Say, for example, your fish produce 40ppm nitrate a week (actually the ammonia that is transformed into the nitrate....but go with the example, ok?). You have 40ppm nitrate at the end of 7 days. You then change out 25% of your water. This change will dilute the nitrate concentration down to 30ppm - for you have removed 1/4 of the nitrates. Now you wait another week. The fish make 40ppm more - and your tank is at 70ppm..... you do a 25% water change and you have about 52ish ppm nitrate left in the tank. LEt it go another week (52+40 = 92) and change 25% and you have 69ish ppm.....
See how the nitrates can climb even though you are changing the water?
To counteract this "creep", you need to do water changes more often or in greater volume such that you can remove ALL the nitrate that is produced each period of time between water changes.
I suggest you do a 100% change. That way you will start (hopefully - if your tap water has no nitrates!) with zero nitrates. Test the nitrates at the end of each day to find out where your levels are going. Then, if you wish to only do a water change every say, 7 days, you can calculate out how many gallons you will need to change to bring that nitrate level down to what it was at the end of the last water change - keeping the levels the same - and hopefully below 40ppm.
(If you already have a fish that is sensitive to nitrates (flipping) I would suggest making an effort to keep the nitrates under at least 20ppm at all times.....)
Does any of this make sense?