Beating yourself up won't help the situation. So put down that stick please.
Over the years I've tried several tactics from punishment which makes life harder to begging.
Neither of those are likely to succeed, they're not healthy approaches. Ultimately you've settled on a path where she hasn't been taught skills and therefore has no confidence to try. That won't help depression.
Instead of viewing her behaviour as "bad" or "a problem" , how about viewing it as a reflection of skills and motivations? Your behaviour has inadvertently driven hers, and hers is inadvertently driving you - but you can slowly start to change that pattern.
She doesn't do laundry because she doesn't know how, doesn't have the confidence, and because she lives in an environment where she receives a desirable outcome if she doesn't do it (because you will then do it).
Think about how you can address that together - involving her in the decisions not imposing them.
Start with a conversation: that it's an important skill for her to have, you want to support her, and you see this as one change she can make that will make her feel better because it gives her control /something to achieve, and you will be there to support with questions and advice but will not be doing it for her.
That kind of thing - not blaming, not criticising, not dictating, but where you want to reach, positive ways you'll get there, and so on. It's a small thing but it will give her a sense of achievement which is very important in rebuilding confidence/self-esteem and recovering from depression.
Pick one goal at a time. This is about changing both of your behaviours not focusing on her as the "problem" or you as the "problem" - there is a dynamic between both of you (neither of you are "problems" you just have some challenging habits to break), but that can be changed.
It will be uncomfortable to start with because change always is, but with repetition and success it will feel comfortable.
Behavioural activation is also a useful strategy for managing depression - essentially that by starting very small and forcing yourself to do activities, your motivation and confidence returns and enables you to do more.
Step away from punishing, begging, blaming (yourself / her). All that will do is damage both of you. Maybe you could look at CBT resources on assertiveness and communication and maybe self-esteem for yourself? Could give you some useful ideas and tactics.