I'm a little confused by your question, but I will try to answer it in general terms.
The company that performed the work may not be entitled to the exact amount that they demanded, but they probably are entited to the "reasonable value of services rendered". If not, then the company that received the work would be "unjustly enriched.".
I don't understand why the company allowed the work to be peformed if it wasn't authorized. Why didn't they stop them before the job was completed, etc.?
A check could be considered a written contract if it showed evidence of a meeting of the minds.