Translation:It took twenty days.
It is correct. When using "x days", the pronounciation is the same as for the ranking numbers ("X-th day of the month"). Full table can be found here: http://www.guidetojapanese.org/learn/complete/date_time
It seems that [はつか] is the priority, besides that [にじゅうにち] is not wrong, it might sound childish. I read it at https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q11156225171
It's the last remaining survival of the Old Japanese word for 20, pata plus the Old Japanese day counter -uka. Old Japanese didn't allow two vowels in a row (unlike Modern Japanese), so pata + -uka became patuka (modern pronunciation hatsuka)
Modern Japanese has borrowed the Chinese numbers, pretty much completely losing everything above 10 except in a handful of vestiges like this.
20 is when you would've been considered an adult. Since it's connected to a special age, it retains the native Japanese pronunciation of はつか/はたち etc. 14 and 24 aren't 'special' so much as that they use the number 4. In Japan, 4 and 9 are unlucky numbers because し「死」 means 'death' and く「苦」 means 'agony'. So, they are changed to native Japanese pronunciations to avoid the dual meanings and therefore 'bad luck'. し becomes よ/よん and く becomes ここの/きゅう. Even the number 7, which is considered lucky, is affected. It goes from しち (which as you guessed, is changed because of the し) to なな.
Of course, it all depends on the context the numbers are used in. Sometimes 7 is しち like when talking about time, and sometimes なな like 7 days / 7th day. Same for the other numbers. It's best to memorize the exceptions.
It's the same for me. I found that I can bypass it by manually changing the number in the URL from a 2 to a 3, and then completing lesson 3 after that got it un-stuck just twice. I'm stuck again, but at least I can still change the URL to skip around and do some of the lessons, even though they don't credit.