'Piacere' can be roughly translated to 'to be pleasing' - if 'she likes the word game', this means 'the word game is pleasing to her', which in Italian is 'Il gioco di parole piace a lei.' But generally, having the indirect pronoun come first is preferred (a lei - le), giving 'to her, is pleasing, the word game'. In Italian, this is 'Le piace il gioco di parole'.
If 'she likes the word games', this actually means 'the word games are pleasing to her' (I giochi di parole piacciono a lei), or reordered, 'to her, are pleasing, the word games'. The subject, most importantly, is the 'word games' (in reality, it is they that are pleasing her, and not she who is 'liking' the word games). This means that, as in the English are pleasing, the verb needs to be in plural form, 'piacciono', giving 'Le piacciono i giochi di parole'.