******************************************
Hi GoldenLyretail:
You'll know the cycle is established when the ammonia and nitrite consistently remain at zero, and the nitrate levels off. In a small tank with currently only one fish the nitrate should be quite low (around 5 ppm).
I would not recommend using distilled water in large quantities. It doesn't have the mineral content needed for fish health and for buffering the ph of the water. I use distilled water to top off the tank when there's been evaporation between water changes, but that's of course a very small amount and it doesn't affect the overall water chemistry (other than slightly diluting the nitrate).
What are your water parameters? Your ph and hardness will be factors in your choice of fish. What kind of filter do you have? The capacity and efficiency of the filter is a factor in deciding how heavily you can stock.
I would begin my getting more corys. They need to be in groups of at least 5 or 6. Once you've added more corys, it would be a good idea to monitor your cycle carefully (testing ammonia, nitrite, and nitrate) to ensure that your biofilter can handle the load.
In a small tank like a 10 gallon you could have a shoal or school of smaller fish like barbs, certain tetras, or rasboras. You could also keep some livebearers like guppies or platys provided you keep the number fairly reasonable. A couple of specimen fish like gouramis would also work in a 10 gallon.
The most important thing to keep in mind is that, in a small tank, changes to water parameters can happen much more quickly (and catastrophically) than in a larger tank.