Is the 'Umrah performed by the wife valid, or is expiation due, if she exited her state of ihram upon reaching Jeddah without completing the 'Umrah due to exhaustion, and did not stipulate a condition when making the intention for ihram, then re-entered ihram from home a week later and performed the 'Umrah?

If the wife comes from Egypt intending 'Umrah, she must perform the ihram from the legal Miqat for the people of Egypt. Performing ihram from home before traveling is disliked but is valid.

Your wife made a mistake by intending to exit her 'Umrah ihram and remains in a state of ihram until she completes her rituals. As for the prohibitions of ihram: whatever was destructive in nature (like cutting hair and trimming nails) requires one fidyah (a sheep to be slaughtered in Makkah, or fasting for three days, or donating three sa' of food to six needy persons). Whatever was for pleasure (like wearing tailored clothes and using perfume) entails nothing if done out of ignorance. And if intercourse occurred out of ignorance, the 'Umrah is not invalidated according to many scholars.