Translation:The kitchens in their houses are small.
At least in US usage, "The kitchens are small in their houses" would be fine. For example, you might say or hear either version in a conversation about XYZ Company, which puts small kitchens in the houses they build, or about some friends who happen to live in houses where the kitchens are small.
I agree, kind of. "The kitchens in their houses are small" (the sample answer) sounds better to me, possibly because "small" on one side of the verb modifies the entire unit on the other side, "The kitchens in their houses". But I do agree that "The kitchens are small in their houses" could also work, as English is flexible with the position of these units of meaning (cf. "The kitchens there are small" and "The kitchens are small there", both OK).
Please check the declension table for "její" (her) at
Notice that some of the plural forms are "jejích", not "jejich".
On the other hand "jejich" does not change when declined and means their.
I do not see either, because "The kitchens are small in their houses." is accepted. As we told you several times, you must, in cases like this one, use "My answer should have been accepted" and you must check for typos just before doing that. More often than not there is some typo or a similar error in the answer.