it's a fixed usage. nalu has different meanings. When it refers to "become" or any verb around it, ni is used. eg. haluni nalu. it becomes spring. bangaosini nalu. she becomes a nurse

びょうき means "sickness" or "state of being unhealthy, ill". に is the particle signifying "in", "a", "into", "to", or "toward". なりました is "became..."、fall into certain state". So it's "I fell in illness", with "I" omitted since it is obvious as in typical Japanese phrases. If you use びょうきが, it won't jibe with なりました, so you must use a different verb. For example, びょうきが きました。(Illness came upon me).

I agree with the two members, びょうきになる is an expression for "to become sick / to be in a situation of not being fine", or "state changes from being fine to not being fine". We use に with なる, for example, わたしはせんせいになります / かれはいしゃ(doctor)になりました / かのじょはげんきになりました, sometimes memorize verb + particle (without thinking too much) is needed. For us, it always happens, for example, "get up, get along with, get on, get off, etc". If I separate these expressions in each words, it doesn't make sence, does it? So, there are so many things that we simply have to memorize when we study language(s). I hope this was helpful.