Hello lanook welcome to the thread.
How old is your daughter? I do think adults expect a lot of kids in terms of accepting new situations, and the "oh they'll soon pick it up" attitude to foreign languages is a bit blaze sometimes - some kids do pick a new language up no problem, seemingly by osmosis, others take longer. If your daughter is quite verbal she might find not being able to communicate verbally with the children around her harder than a more physical, less talkative child. Either way it'll come of course but you can see why she'd have wobbles after the initial novelty of the first week wore off!
I can't directly help with the non German speaking child entering Kindergarten wither though, sorry! My kids are nearly 6, nearly 4 and a baby, so we have been doing the kindergarten thing for the past 3 years already, but we moved here when my eldest was 19 months and the others were born here, so by the time my eldest started KiGa at 3 her German was not bad - in the year and a half before Kindergarten we did Mütter-kind groups (toddlers) and made an effort to get to know as many children as possible in the local playground, plus even though DH did not talk German to her (another story) he is actually German so she had contact with his German speaking relatives, which all helped.
My German was pretty much non existent when we moved here and is still a bit rubbish, but I talk to anyone and everyone and we live in an area with few willing English speakers, so I have now reached a hodge-podge stage where I understand most things and can say what I need/ want and be understood, but I sound ludicrous due to dreadful grammatical structures and accent, and people have to be patient with me - most are luckily!
My only advice is throw yourself in and try to speak German and it will come, though formal study is also a good idea so you don't sound like a mad woman like me :) For your daughter I guess you just have to view it in the same way as if she was having wobbles starting any new childcare situation regardless of language - even if you'd moved to Australia say, the same might have happened with wobbles after the first week in a new nursery/ preschool etc.
If she is a bit older though you could ask the KiGa about getting her some extra help - a 3 year old and a 5 year old are very different cases when it comes to suddenly being immersed in a new language in my opinion... Kindergarten may be able to access a German as an additional language programme, but these are often for the older Kindergarten children approaching school age.
All the best, sorry not to have been much help.