You are right to say that it is not a complete sentence but it does have a verb. 'strick' is the verb in the clause.
To make it a complete sentence, you can change 'for expressing' to 'expresses'. We normally use 'among' instead of 'between' when we talk about more than two things.
The card expresses your heartfelt feelings in a way that strikes the right balance among adorable, sincere and tasteful.