[John] Kate, could you make sandwiches for the whole family please?
Kate needs addressing directly.
[Kate] Sure, I am happy to do this.
The construction is fine as it stands, though I doubt any native would get past just "Sure"
If they did, I'd think they would be more likely to say
Sure, I'd be happy to do that
'I would be' indicating the task isn't yet started, which pushes 'this' towards 'that'... at a distance, measured in time rather than metres.
"I'm happy to do this" more implies they are currently making the sandwiches.