'Luckily' fits well here, or 'fortunately'. So does 'by chance' if you moved it: 'by chance I got' (and the other two fit there too).
I don't want to say that 'accidentally' or 'by accident' mean there was a real accident (an event affecting someone badly), but they do imply something more, so that they're not quite suitable here. Perhaps they refer to something that could have gone wrong: I accidentally went to the wrong bicycle shop, and that's how I met my wife. (It turned out lucky, but might have caused me nuisance or harm of some kind.)
If you were hiring a £10 bicycle, and the man at the shop accidentally gave you a £20 one, you would luckily get a better bike, but it would be by accident, not by (pure) chance. Getting a green one or a blue one would be by chance, but not by accident (assuming you didn't ask for a particular colour).
Entangledbank, thanks. Can I sum up your explanations that if something happens 'by chance' it is a random incoincidence while 'by accident' is caused by an unintentional fault of a person (and as you explained with possible bad consequences)?
In case of your example for 'by chance' with the bycicles where is the bad consequence? Is it that the man at the shop actually became £10 short?