The simplest methods are usually the best. Try this code

 N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15]//InputForm

which returns

 4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

 4.5899999999999999999999988020950000000000000000001613`34.*^-25