Welcome to the forum.
If you are following the recipe indicated, that is, the Tyler Florence recipe, then you are using 00 flour, which is an imported Italian flour. I saw the Foodnetwork segment in which Florence made the dough and a few pizzas, but I couldn't make out (or don't recall) the brand of 00 flour he was using. Do you know what brand of 00 flour you have been using? This question is important because 00 flour comes in different amounts of protein, from about 8.5% to 11.5-12.5%, which can affect kneadiing. I know of only about 4 brands of 00 flour that are available at the retail level (including a King Arthur "clone" that I do not personally recommend).
The classic Neapolitan dough recipe using 00 flour usually calls for a total of around 30 minutes machine kneading and 2 rise times, a first one of about 4 hours at room temperature and a second one of about 2-4 hours. I have kneaded 00 flour dough by hand on many occasions. As best I can tell, the reason for the long knead time is to develop the gluten, which is formed in relatively small quantities when compared with higher-protein, higher-gluten flours. Also, my experience is that with 00 flour you will not get a really good gluten windowpane, apparently because of the smaller amount of gluten--which is what creates the gluten window. I would just knead the dough ball until it is smooth and elastic. I would think that about 10 minutes should do it based on my own experience. And I echo Steve's advice about being sure to use enough water, since that is likely to help you overcome toughness of the dough and get an airy crumb in the crust. The dough shouldn't be so sticky that when you put your fingers in the dough and pull them out the dough pulls out with your fingers in long sticky strands that you have to pull off. I would add just enough flour to prevent that from happening and maybe just a bit more. I think then you should be OK.
When I reread the Florence recipe I saw that the recipe calls for 12-15 minutes bake time. That seems overly long to me for a dough based on 00 flour and 1/8-inch thick. Since the Florence recipe calls for both added sugar and olive oil (neither of which is used in classic Neapolitan doughs), you should get good browning but you will have to check the bottom of the crust as soon as you start to see browning on the top crust to make sure that you don't overbake the pizza. For pizzas baked on a pizza stone or tiles, the typical oven temperature is 500-550 degrees F, which is the highest oven temperature for most home ovens. The usual recommendation is to preheat the stone for about 1 hour before baking a pizza on it.
You indicate that you don't have a mixer. Do you have a food processor by any chance? That is a good alternative although the processing techniques differ markedly from a stand mixer. It will also improve the quality of your dough.
By and large, it sounds like your problem is mainly a problem with getting enough water in your dough and kneading it enough to get good gluten development. This will usually be more of a problem with hand kneading higher gluten flours, such as bread flour (including bread flour for bread machines) and especially high-gluten flour which is a real bear to knead by hand, especially if you are making a large quantity for several pizzas. If you are using a higher gluten flour, you should also be certain that you are using the right recipe. You can't just substitute such flours for 00 flour or all-purpose flours in recipes.