In particular, I don't understand what is the purpose of どうすれば and why こと appears two times.
My obviously bad understanding is:
What important things you can properly convey through the phone...
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What should we do to make sure we properly communicate the necessary things over the phone? (loose translation)
Note the お -> を typo you had.
Regarding どうすれば I can do no better than point you to this link. It uses どうしたら but the meaning is the same. An in-place translation of どうすれば might be "by means of doing what ...", but there is no natural direct translation.
Regarding the two ことs, the first one means 'things'. 必要なこと = "things which are needed". The second こと is part of the usual pattern "verb + ことができる" which means "to be able to do verb".
A word-for-word translation might be something like "by means of doing what (どうすれば), by means of phone (電話で), necessary things (必要なこと) properly (きちんと) can we communicate (伝えることができるか)". Which roughly becomes my initial loose translation.
I should also note that using か to indicate a question after the plain form of a verb is a little rough. I wonder where this sentence came from?