n=355 , and you will need to flip the coin about 3550 times
So looks like winning 10 even bets in a row is about 1 in 3550 and I have 104 tries
That's completely wrong and I don't think there's any need to bring quite advanced maths into it. That maths refers to how many bets you'd need to do before you were 99.99% sure you'd have 10 winning bets in a row. And that's 10 bets in a row starting from any bet, not just from the £10 one so it is quite irrelevant to what you're doing.
A logical way to think about it is if you are getting 100% value on your bets, and you were to turn £10 into £10240, then the odds of doing it is 1239/1 for every £10 you start with. With 104 tries, your odds on getting to the £10240 target come down to about 12/1 or about 8%.