First, the stoichiometric equation for the reaction between NH3(aq) and HCL(aq) is:
NH4OH + HCL -> NH4Cl + H2O
It shouldn't actually be written as NH4OH, but when you dissolve NH3 in water, you get NH4+ and OH- ions.
Anyway, since your concentration of HCL was 0.135 M (this means 0.135 mol dm-3, which means 0.135 mol in every 1000cm3, or 1 litre), and your volume of HCL used was 9.70mL, then the amount of HCL was 0.135 x 9.70/1000 = 0.0013095 mol.
So your amount of NH3 in that 10 mL was also 0.0013095 mol, since NH3:HCL is 1.1.
You took the 10mL of diluted NH3 from 100cm3 of stock solution, so the stock solution contains 0.0013095x10 = 0.013095 mol of NH3.
Now, this is the exact same amount of NH3 present in your 20 mL of UNdiluted NH3, since you just added water but the amount of NH3 remains the same.
So that means that 20mL of your original cleaning solution contained 0.013095 mol of NH3.
And hence your cleaning solution's concentration is 0.013095 x 1000 / 20 = 0.65475M.
I think this is correct (can someone check, thanks), and I noticed that your calculations did not divide the volumes by 1000, which you must do since they are in mL and M means mol per 1000mL.
So, to answer that last question, yes you have to take into account dilution if you want to find out the concentration of your original cleaning solution. But if you just want to find the concentration of the diluted NH3, then it would be (.135 x 9.70/1000)/(10/1000). Which is which Borek wanted to know