WElcome to goldfish keeping! It really is not all that stressful and hard - but, as you have noted, getting started can sometimes be a pain.....
First of all - I am soooo sorry about your moor. THose are such lovely fish. With a 10 gallon tank, though, you only have enough room for one goldie - and if that one grows, you may need to invest in a 20 gallon or larger for him.
Lets examine everything here, OK?
First - your filter. You know about the nitrogen cycle - that is great. It does take time to develop - 8-10 weeks sometimes. ACK!

I know. But when it finally kicks in, it will be such a relief that you will LOVE it. Until it does, though, you are going to have to diligently work at keeping your tank liveable for your fish.
To do this, you are going to have to do many more water changes of much larger volume at a time. Your goal is to get and keep the ammonia in the tank at 0.5ppm or less. I prefer 0.25ppm myself.
If you are getting 2ppm ammonia and change out only 1/4 of the water that is still 1.50ppm ammonia. Still too high. If you change out 1/2 the water, you STILL have 1ppm ammonia! STILL to high. So, you will find, at least for a time, you are going to have to change out at LEAST 75% of the water. I think I would suggest that the first time here to get things back in order, you will take out 7 1/2 gallons and replace it. Then immediately, take out another 5 gallons and replace it. This will get 2ppm ammonia down to about 0.25ppm ammonia. Now you are in the range where you want it to stay, more or less.
Cutting back a bit on how much you feed your fish will go a long way to limiting the amoount of ammonia. Smaller meals or feeding only every other day or so, during cycling may actually help your fish - for you can control the water better that way.
So, now, everyday, you should test your water. If the ammonia is over 0.25ppm, change out enough water to get it back to the 0.25ppm. That may be 5 gallons or 7.5 gallons. Do what is necessary. Do not worry - the developing biocycle is NOT in the water, it is in the filter.
Adding the Amquel+ to the water will detoxify any chlorine as well as bind any ammonia/nitrite and even a bit of nitrate from the water. This product (Prime is another one) does a pretty good job of this and is an invaluable tool when cycling a tank. It chemically binds the ammonia from the water, creating a different molecule. The beneficial bacteria that make up your biocycle can still peel the ammonia from this molecule and utilize it in their life cycle, but the ammonia is not free in the water to bother the fish. GREAT, huh?!
Adding distilled water will work. Goldies can, and do live in higher pH values very easily. 8.5pH is a bit on the high side, but, aside from a few medications that are more toxic or less effective in higher pHs, the goldies should be able to adjust and do ok in that pH if you wish. I would like you to test this pH, though, by setting out a bucket of tap water overnight or for a day. Test the pH out of the tap, and the pH at the end of the time period. Comparisons of both will help - the pH may drop over time. When you have a bang-up biofilter working for you, the action of the bacteria breaking down the ammonia/nitrites releases acid and tends to naturally lower pH, also.
If you have the ability, it would be very nice to know what your kH/gH are. Distilled water is totally void of anything other than H2O. Next up on the scale is DI water and finally, RO water. RO water has had the majority of the mineral removed, but still has a bit of stuff in it. Those minerals are VERY important for the fish to have - they need them to live and thrive. This is why you should never use all distilled/DI/RO water. Since you are mixing it you should be getting enough in the water. But testing would not hurt, just to make sure. Realize that RO/distilled water has a pH of 7.0 but ZERO buffer (kH) It will not hold that pH. It will bring down your pH overall, but may not keep it there, depending on your kH in the tap water.
If you wish to mix the waters, that is fine. I mix RO with a bit of filtered water to get what I want. I also add in powdered minerals and calcium to bring the gH and kH back to where I want it. Once you get the formula for your water, you can continue to mix it up the same way each time. When you get good at it, you can test only at the end of the production just to make sure.....
Adding Amquel+ can be done two ways. You can add the dose for the entire tank directly to the tank and then add the new water. Or, you can pretreat the new water and then add it to the tank. Amquel is a very forgiving chemical. It will bind what ammonia/nitrites/nitrates are in the water, but, if nothing is there, it simply waits, benignly, to be needed. It is tough to overdose. It pretty much works instantly - as soon as it is stirred in, you are good to go.
Lets break this down more simply, though....
1. Change out LARGER volumes of water. Keep the ammonia and soon, the nitrItes, down to UNDER 0.5ppm. Aim for 0.25ppm if you can. Do not worry, the cycle will develop just fine and you are not bothering it. The bacteria are in the filter, not the water.
2. Add AmQuel+ to the water. You can add the dose for 10 gallons directly to the tank and then pour in the new water, or you can treat the new water and add it to the tank. While you are cycling, the AMquel+ is NOT just for tap water conditioning. It is BINDING the ammonia/nitrites from your water making it less toxic for the fish. It is an invaluable tool in protecting your fish. It will keep the ammonia/nitrite under control between your water changes.
3. You can add salt to the water - to a salinity of 0.1%. That is 2 level Tablespoons of large grained aquarium salt for your 10 gallon tank. Dissolve it in a cup before hand and pour it into the tank. Do not pour it into the filter. Do not pour it onto your fish. This will be particularly helpful in protecting your fish when you start to get nitrItes.
4. Have PATIENCE. A cycle really will happen. It just seems to take FOREVER! But it really works. The nitrites will be particularly frustrating, but they DO get processed.
5. While you are waiting, change change change the water. And come on back here and share your adventures, questions, comments and good humor with us all.