行き場のない気持ち を英語に訳すと？

すみません、her mother を his mother に訂正します。
The son’s feeling of the unmitigated stress which is a result of providing excretion care for his mother and living together with her over an extended period of time.

行き場のない気持ち
a stressful emotion that cannot be relieved (mitigated, alleviated)

息子が母親の排泄ケアやずっと2人で同居し続けることに対する、どうしようもないストレスフルなきもち
The son’s feeling of the unmitigated stress which is a result of providing excretion care for her mother and living together with her over an extended period of time.

長いですが、これは文ではないことに注意してください。元の日本語のように「気持ち」の修飾語が長いだけです。