Luckily, Washu detected a subspace disturbance when KAIN first escaped and went back in time. Becuase of this, she was able to save Tenchi from vanishing, for a while. While Tenchi was slowly fading away, the Tenchi crew had to go back in time and to save Achika.

When bad got to worse Achika finally realized her powers and was able to stop KAIN. Washu sent in a bomb into subspace and destryed KAIN for good. Unfortunately, Achika still died when Tenchi was about five. The rumor is that she used so much of her power, that she shortened her lifespan. She might have lived on if she could have remembered her powers. (Or came by them inherently.) Before the Tenchi crew left Achika's time line, Wahsu made it so that Achika could not remember anything that had happened.

How Achika Died......

There are few different theories on how Achika died, There is her death in the TV series, the OAV, movie and novel version. I have now thoroughly researched this. This is the best it gets folks. And don't let these other rookies fool you, their info kinda looks like mine.

Tenchi Muyo! - In Love (The Movie)
It seems that the cause of Achika's death is her over-expenditure of power, thus reducing her lifespan. (Remember the king?? he killed himself in trapping Kain.) Also, I've read on the internet that being that close to the firing of a dimensional cannon, may have some side effects. Also, having come in contact with people from the future, could have done something, and maybe, just maybe, if they would have left her memory of the incident with Kain, she could have avoided some unseen event causing her death. By understanding her true power and being able to use and control it, could have given her the power to stay alive, unfortunately that's not the case.

Tenchi Muyo! OAV TV Series & Novel
I guess the series runs off the information given in the novel in which the movie is made from. Kain, also did not make an appearance in the OAV & TV Series. Poor Achika died of a broken heart, even though showing great promise as the next lineage of the Jurai royal family, and having great powers as well. Achika, being the bright beautiful girl she is, fell in love with a local villager and wanted to be a normal girl. Although she did not want to disappoint Yosho as well, though her true feelings came through, and not allowing her to inherit the true Jurai powers, one being the royal tree Funaho. It was during the ceremony with Funaho (royal tree) that she incurred an injury(remember tenchi oav #5.) During her recovery, the man she loved betrayed her. She recovered, even though very weak, and married another man, Nobiyuki, whom she seemed very happy with. She thought he was a spitting image of Yosho. A year later Tenchi was born, 4 years later she died.

If you have any info, feel free to e-mail ME