if your tank is an empty tank then there is no need to run the chiller during cycling, true although the evaporation rate would increase but the speed of bacteria build up will be slower, so to hasten the process (i assume you want it to be faster) the chiller being off would benefit more.
high nitrite shows that one portion of the cycle is complete.
you do not want to reduce the nitrite in order to be able to continue cycling.
different types of bacteria convert the different elements into nitrite and nitrate respectively.
nitrosomas--> ammonia into nitrite
nitrobacter--> nitrite into nitrate
thus you will need nitrite to complete the cycle at least until the nitrate stage.
you can theoratically further speed up the process by purchasing off shelf bacteria, however best is to leave the tank to its own devices and increase water contact with atmosphere, via airstone, etc,
you will need places for bacteria to establish, be it, rocks, sand, bio-balls, bio-homme or even filter wool. after cycle is complete the bacteria will be residing in those materials.
the bacteria will build up accordingly, waiting time about 2 weeks to a month depending on different tank situations. when ammonia and nitrite is 0 cycle is complete, nitrate will be higher and a significant water change can be performed. Subsequently fish can be added in at a recommended rate of one fish per week.