私もその妻として恥ずかしくないよう、微力を尽くします。 As his wife, I will ***
I don't understand the end at all. What my dictionnary says and what the sentence says doesn't seem to match.
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A somewhat loose translation of the sentence as a whole would go something like,
"I too will do what little I can, so as to live up to my position as his wife."
微力 is a humble expression referring to "one's meagre/limited abilities". 尽くす in this case means "to use something to their fullest extent". So when put together, 微力を尽くす is like saying "I'm not capable of much, but I'll do everything I can with what abilities I do have."
The 妻として恥ずかしくないよう is also something of a set expression; if you are [position]として恥ずかしい it means you are an embarrassment to that position; ie. you're not living up to what is expected of someone in that position. So 妻として恥ずかしくないよう indicates that she's going to put in this effort in order to avoid being "an embarrassment as his wife", the implication being that she thinks a lot of her husband and feels she needs to put in an effort to be a wife befitting of his status.