高兴 describes a mood or emotional state. For 'willing to,' it often appears in the negative: 不高兴 can mean 'unwilling' or 'unhappy' depending on context.
mistakes
Don't confuse 高兴 (happy, a feeling) with 开心 (happy, more casual/colloquial). They overlap heavily, but 高兴 is more often used in polite or formal phrases like 很高兴认识你.
grammar
认识你很高兴 is a set greeting. In everyday speech you'll also hear 很高兴认识你 - both word orders are natural.
