There are a couple of things working against you here, most likely. As you found, and Koko advised, your tap water itself has nitrates in it. So you are never, ever going to be able to get lower nitrate readings than 5-10 ppm nitrates.
Secondly, when you do a water change you need to actually look at the dilution that you are doing. For example:
If your tank water has 40ppm nitrates and you change out 25% of the water, you would be lowering the nitrates to 30ppm. But since you are adding back in water that already contains nitrate to the tune of 5-10 ppm, you are probably going to only lower the overall quantity to approx. 34ppm. In a drop test kit, the difference between 40ppm and 34ppm in not possible to see. So it seems as if nothing changes. You also have to remember that the fish are constantly adding more ammonia that is constantly being converted to nitrate - a continuous source.
Since you cannot change your tap water, you should, perhaps, change your water changing habits. I would suggest that you start doing 50% water changes each week - up to 75% water changes if necessary. As long as your beneficial bacteria is contained in your filter well, and you have a solid platform that you do not disturb, you will not bump or even nudge your cycle by changing large amounts of water. (IF you were to change 100% you would still have 5-10 ppm).
It is possible to help reduce nitrates by eliminating all possible pockets of waste that may or may not be properly cleaned each week. If you have a cannister, you need to empty out the reservoir of water in the bottom of this. Leave the bacterial platforms alone, but clean the yuck out of the bottom of the filter. IF you have gravel, reducing the amount or eliminating it can also help to a degree.
I think if you simply up the volume of water that you change each time, and perhaps decrease the times between changes you will get things back into shape.