I believe that it is fairly straightforward to do the paperwork yourself, but I haven't personally done it, so perhaps someone else can advise on this.
If the helper is finishing her contract, then, as you say, she would not normally have to leave HK while her contract with you is processed.
If you do use an agency, then please ask them (and ask your helper) whether they would charge the helper a placement fee. Under HK law, a HK agency is not allowed to charge the helper more than 10% of the first month's salary (and this is not payable until after that first month's salary has been received). In the Philippines, it is illegal to charge any placement fee at all (if, indeed, a Philippine agency would need to be involved in this case). I have heard that Emry's do not overcharge helpers, but it would be as well to check!