If not how would it be corrected using the exact same characters?
And what is the issue with the original sentence?
Struggling a bit with word order!
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If the rule of the game is all characters must be used, I guess I can play it like this:
"打扫这么大的客厅多累啊!" (cleaning such a big living room, how tiring!)
what is the issue with the original sentence?
"打扫大客厅"(cleaning the main living room) is the topic, followed by the comment/opinion "多累啊"
'这么' and '的' are not needed
The issue with your original one 打扫大客厅这么多累的啊 is that you put two modifiers 这么 and 多 for 累，so you might have to remove one. And also 的 is misused and should be removed.
Also, 这么多（的） + noun is the correct structure, like 这么多（的）爱，这么多（的）人. However, 这么多（的) can not modify an adjective, so 这么多（的）累 is a wrong expression.