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The only one I can come up with is 「そうなんですか！」 but even then the meaning and
feeling is slightly different than "I understand/I got it".

Context:  
So if my Japanese teacher teaches me something useful, I'd say something like
"Ah! I get it now.", but I think using 「分かりました」 all the time makes me sound
like a soldier.
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phrases
Are there any similar words or phrases for「分かりました」
そうなんですか！ means something like : "is that so!?"

You could use the following 4. Which are used quite frequently.

> 承知{しょうち}しました  
>  了解{りょうかい}しました  
>  了承{りょうしょう}しました  
>  かしこまりました

They all have about the same meaning, but are used in different occasion
mostly depending on required level of politeness.
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I think you could say...

「なるほど！」  
「あぁ、そうなんですね！」  
「あぁ、そうか、わかりました！」  
「あぁ、そういうことなんですね！」(← This is like "Oh, that makes sense!")

* * *

I think it would be alright to repeat わかりました though... Maybe you could use
some fillers, like
「あっ、そうか、わかりました（、ありがとうございます）！」...「あぁ、なるほど！わかりました！」...「あぁ、そうなんですね！わかりました。」...
then you wouldn't sound so repetitive.
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