Kakashi's Sharingan had indeed remained activated in its Three-Tomoe state the entire time. Keeping his eyes closed was completely useless. Objectively speaking, activating the Sharingan was a combat-ready state, and it inevitably caused a massive loss of Chakra. Even members of the Uchiha clan would not do this. Kakashi covered his left eye with his forehead protector, perhaps as a subconscious attempt to "plug the leak," but in reality, this kind of "physical blockage" was completely useless.
As for Kushina, using Fuinjutsu to cut off the Chakra supply to a certain part of the human body was naturally no difficult task. Even cutting off the Chakra supply to the eyes... would merely require more precise control.
This kind of Fuinjutsu was essentially no different from Yuhara's Six Dimensions Seal. The only difference was that the seal usually remained closed, and would be opened when the Sharingan was needed. It could even add an adjustable "valve" to control how much Chakra reached Kakashi's Sharingan.
After hearing Yuhara's explanation of the problem with Kakashi's body, Kushina immediately began taking it seriously. No wonder she had always felt that Kakashi's strength should have been growing like an "inflating balloon," yet had barely increased. As it turned out, he had been leaking air the whole time.
As for Kakashi never taking the initiative to mention his problem to Kushina, that was actually because he had not cared about such a minor issue before... After his teammates died, especially after he failed to protect Rin and even let her die by his own hand, saying that Kakashi had given up on himself might be an exaggeration, but at the very least, he had simply let himself drift.
Kakashi's experiences were somewhat better now than they had been originally. At least Kushina was still alive, and