Let’s look at the pieces:
"I've been staying with the Shinseki family/visiting Shinseki lately, and I really feel the need to improve my English skills. Being here, surrounded by new experiences and meeting new people, makes me realize how important it is to communicate effectively in English. Therefore, I've decided to dedicate some time each day to studying English. I believe that getting better at English will not only help me during my stay here at Shinseki's but also in my future endeavors."
Is “shinseki no ko to o tomari da kara eng better” correct English or Japanese? No. Is it correct communication ? Absolutely.
of an anime often associated with this trend, or are you looking for translations of specific lyrics?