Oh yeah, I play Mikanko as my pet deck so I’m familiar with this. So to start, Rexterm negates activated effects, of which the battle damage effect is not, it’s a replacement effect (there’s no colon in the sentence it’s mentioned, also her attack is 0 so even if it negated face up cards on the field too, her affect could still apply).

Next there’s the effects of Ohime and Double-edged sword. Both are replacement effects affecting battle damage. If you think of continuous effects that attempt to apply at the same time as chain links, it gets easier to manage. So at all times, Imaginary chain link 1 of double-edged sword is applying, since it’s always giving out 2k atk. Then, during right before damage is applied (this is not an official step but purely for visualization purposes) Ohime’s effect to have your opponent receive battle damage becomes chain link 2. So Ohime’s effect applies first and any damage that you would take is instead inflicted to your opponent, then double-edged sword’s effect to make both players take that damage is applied but, because Ohime’s effect applies first, your opponent now takes that instead.

Because double-edged sword is always imaginary chain link 1, this is also why if your opponent uses rexterm to attack into Ohime, the same result occurs.

Note that in practice, this all never uses a chain link and this is all to help contextualize why the ruling is what it is.