So...the scenario is this:
1. Person X sends a demand letter to Person Y. The letter states that, if Y doesn't respond by mm/dd/yy, a lawsuit will be filed. We have no more information about what the letter says.
2. Y did not respond by mm/dd/yy.
3. Despite the threat in the letter, X has not yet filed a lawsuit (which makes your references to "the defendant" a bit confusing).
Is that accurate?
And your question is whether Y is entitled to recover for some sort of attorneys' fees. Under the facts given, there is no apparent reason why Y should have incurred any attorneys' fees, and there is certainly no obligation owed by X to Y or ability for Y to recover anything from X. Note that, even if X had sued Y and Y prevailed, Y would not necessarily be able to recover his/her attorneys' fees. The "American rule" is that each side pays its own attorneys' fees. The exceptions are for suits based on a contract that provides for the recovery of fees and suits based on laws that allow for it.