My Japanese is a bit rusty, but I'll try to help out.
Firstly, I just listened to the only-conversation track, and it seems that Sayaka-san says "ga", not "wo".
Whether you use 'ga' or 'wo' depends on the type of verb being used. With verbs of ability, 'ga' is used to mark the object, while otherwise 'wo' is used to mark the object.
'Ga' is more often a subject marker, but in this case, it functions like an object marker instead of 'wo'.
Unten ga dekiru : I can drive
Muzukashii buzurigaku no shukudai ga kakemasu : I can write difficult physics homework.
Whereas for normal non-potential form verbs, 'wo' is used.
Chuugoku-go wo hanashimasu : (I) can speak Chinese
Yamada-san wa raishuu no kaigi no yotei wo shitte iru. : Mr. Yamada knows the schedule for next week's meetings.
If this seems a bit arbitrary, it might help to think that in cases with potential verbs, 'ga' is marking the subject of the sentence.
'Watashi wa Eigo ga hanasemasu' might be though of as "As for me, English can be understood". (although hanaseru isn't passive, so maybe don't use this method).
Another thing to note is the Ga can be used with verbs of ability (dekiru, wakaru etc.) which aren't in their potential form.
i.e. Eigo ga wakarimasu and Eigo ga wakaremasu might be used interchangeably.
If there's something horribly wrong with my explanation, please correct me.
Off topic: I can't wait for my exams to finish next month so I can study more than 30 mins of Japanese a day :)