Here are some clues that might help you understand the sentences. I'll write these as questions. Can you figure out how to answer these questions?
どうして私は健康に注意するようになったか。
日本の世界地図はどのようにかかれているか
この小説家はどんな小説をたくさん書いている。
Are you able to translate these questions into good English? If you are and are comfortable with that. Then you're on the way to making sense of the original sentences.
What makes the first sentence more of a challenge to translate is the phrase をきっかけとして.
きっかけ means an opportunity, the occasion for something to begin, an excuse. It's a very flexible word in Japanese and to translate it into English requires a bit of reflection.
In this context, I would render 入院をきっかけとして as "having been admitted to the hospital became the opportunity for ..."
It's not a particularly good translation because I'm inverting the word order and which part of the sentence is the main clause and which the subordinate clause. But some things are just said differently in Japanese.
For example, in English it is quite natural to say
Lots of Japanese like okonomiyaki.
And as a beginning Japanese student who speaks and thinks in English, you are likely to translate this into Japanese as follows:
たくさんの日本人はお好み焼きが好きです。
You'll be understood. But it isn't very natural sounding. This sort of expression sounds more natural as
お好み焼きが好きな日本人は多いいです。
I think it is useful to think of きっかけ this way too. In other words, you might want to flip the order of what's what from Japanese to English.
A couple other things to point out. 中心 here definitely does not mean concentration. 中心 just means center.
The other thing I think I see you doing is rushing in too quickly to decide what the object of the sentence is etc.
I would recommend you start from the main verb. Since that's always at the end of the sentence it is easy to identify. Pay attention to the verb. Is it active or passive? For example かかれている is a passive-form derived from the verb かく. And, in the case of maps, you draw a map. So, かかれている means "something is drawn".
Also, in noticing the verb, check to see whether the real verb is somehow more deeply embedded. In the first sentence it is not very useful to just notice なった as the main verb. The verbing is being done with 注意するようになった. And it'll be important to recognize that this is a kind of set phrase to express the idea that a habit (in this case) that hadn't previously been established began to be developed.
Next, what is the subject and object for the verb (if there is even an object). Don't latch onto the first が or を you see. Those might be embedded in a relative or subordinate clause. And keep in mind that は only marks the topic, but grammatically the topic can be the subject or object or just be the perspective from which the sentence is expressed.