High nitrate is normal in fishless cycling. Every ppm of ammonia you've been adding gets turned first into 2.7ppm nitrite, and that gets turned into 3.6ppm nitrate, so every 5ppm ammonia you add ends up as 18ppm nitrate. It can build up pretty quick. You can end up with a very high level of nitrate, this is why you will need to do an enormous water change before you get fish, to get rid of all the nitrate.
Look at it this way, if your nitrate is going up, there must be some nitrite eating bacteria growing in the filter.
Yes, keep adding ammonia at 24 hour intervals now that the reading is dropping to zero. But add a bit less, dose to maybe 2ppm until your nitrite drops to zero, then increase back to 5ppm ammonia till both ammonia and nitrite drop to zero in 12 hours. That'll stop the nitrite level getting too high till the nitrite eating bacteria have grown some more. Once they are processing the nitrite made from that lower amount of ammonia, you can increase the amount of ammonia you add and it won't take them long to get rid of the extra nitrite that'll be made.