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I'm sorry I don't know what your question is referring to. Both the Greek and the English sentences seem correct as shown at the top of this page.
is there any method to figuring out the past tense or do they all just need to memorized?
The continuous past is easy: take the present-tense stem, add a different set of endings (-α -ες -ε -αμε -ατε -αν), and move the accent to the third syllable from the end -- or to an accented έ- if there aren't enough syllables.
The one-time past is a bit more difficult: you need the aorist stem for that, and the relationship of present stem to aorist stem is not always simple.
It often involves adding a -σ-, but sometimes there are additional changes.
And for a few verbs, both stems are the same, as with κάνω, so έκανα is both "I was doing; I did (regularly)" and "I did (once)".
The good news is that once you've figured out the aorist stem for the past, you'll also use that for the future and for subordinate clauses. (Except for a tiny handful of exceptions where the aorist stem in the past and the aorist stem for future and subordinate clauses still look very different, e.g. είδα, είπα, πήγα versus δω, πω, πάω.)
So the basic principles still work like in Ancient Greek?
How do you identify when to use "τι" vs. "οτι" in a context like this?
οτι doesn't exist as a word in Greek.
ότι wouldn't make a lot of sense here to me: ξέρω ότι έκανες would mean "I know that you made" or "I know that you did", but using κάνω without an object sounds odd to me. What might make sense is ξέρω ότι το έκανες "I know that you did it".
ό,τι (the comma is part of the word) means something like "whatever" or "everything that": ξέρω ό,τι έκανες would mean "I know everything that you did".
Or if we use a word that does not need, but can take, an object:
- Ξέρω τι έφαγες. = I know what you ate.
- Ξέρω ότι έφαγες. = I know that you ate.
- Ξέρω ό,τι έφαγες. = I know everything that you ate.
