O que significa 疲れるものは疲れる ?

See a translation

It means, 'It is ---, anyway.'
In the case of your question, the phrase that came earlier says「いかに慣れているとは言っても」would be the key.
It says 'Even though he was used to dealing with such matters' and comes to 「疲れるものは疲れるのだった」which means 'what is tiring is tiring, anyway'
So, altogether it says, 'He has always been taking care of the two and has gotten used to it now, but still it is tiring anyway, after all.'
In the pattern '---ものは----,' usually adjectives come to the blank part ----.
Some examples are:
A ほうれんそうは嫌いだよ　I don't like spinach.
B でも，バター炒めにすると，おいしいよ？ Well, it is good when sauteed.
A 嫌いなものは嫌いなんだよ。。。。I don't like spinach, anyway.

A 宿題，やりたくないなあ。。I don't wanna do homework.
B やってしまえばすっきりするよ。コーヒー入れてあげるからさ。You'll feel better once you are finished with it. I'll make some coffee for you.
A 何と言われても，やりたくないものは やりたくないよ。I don't want to do it anyway, whatever you say.

A あんまりケーキばかり食べてると太るよ You'll get wait if you keep eating cakes the way you do...
B 食べたいものは食べたいの！ I want to eat them anyway!!
などなど。