"My wife was in Beijing last week."
The chinese grammar Wiki says that zai 在 cannot use the aspect particle 了. Also, by definition, 了 is used to signify the verb is a completed action. But 在 is not an action but a state that is not ever completed, only changed. So it doesn't fit the logic. And in this sentence, you don't need it because the time is indicated by the time phrase 上个星期 last week.
Your sentence would probably be disqualified because you said 我妻子 which is quite common but seems to be consistently rejected by Duolingo. Recommend that you always add the 的 while on Duolingo: 我的妻子.
The main point of your comment, though, is about 去了 vs 在. 去了indicates that your wife travelled to Beijing last week, that is to say, she completed her trip from somewhere to Beijing. 在 simply indicates that she was there; she could have been there for the whole week, she might have been there for a year, she might still be there now, or she might have been there for only a day.
I don't think the English implies anything about where your wife was prior to, or since, last week.