I am surprised they didnt provide you hotel vouchers coz this is what they usually do - and when Lufthansa provides hotel the hotel depends on your class of travel and status. They are also required to provide assistance as per EU261 and book an accommodation for you. Or did you just decide yourself to go straight to the hotel?
In case of booking yourself - while there is no official cap - they are only required to compensate on what is reasonable. Hence in case there are hotels within that budget, then one could argue that that limit is reasonable and especially if you fly Economy there is no need to book you in 5* hotels. There are also multiple hotels around the airport - so the taxi ride would not be that far