I'm confused. You write your scenario in Present tense verb forms:
"...her father who has dementia."
"...what her father is thinking."


...yet your dialogue is in the Past tense, as if he is dead, or too incapacitated now to follow a routine:
"...one thing he had left..."
"...when all his memories were fading away"
"...Dad knew that..."


As per the tenses used, my version would be:
His routine, the one thing he had left that he could count on, kept him hanging on to something solid
His routine, the one thing he had left that he could count on, let him hang on to something solid

when all his memories were fading away. Dad knew that,
“First I do this, then I do that, then I go here, then I stay for awhile, and then I go there.”
“First I do this, then I do that, then I go here, then I stay awhile;((semi-colon))) and then I go there.”
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With suggested tenses, my version would be:
His routine, the one thing he had left that he could count on, kept him hanging on to something solid
His routine, the one thing he has left that he can count on, lets him hang on to something solid

when all his memories were fading away. Dad knew that,
when all his memories are fading away. Dad knows that,
“First I do this, then I do that, then I go here, then I stay for awhile, and then I go there.”
“First I do this, then I do that, then I go here, then I stay awhile;((semi-colon))) and then I go there.”