@PrinceNaveen
I know someone going through similiar at the moment. A strategy that works for him is to write a list of priorities and start working through them one at at time. Try to make sure you do at least 1 thing off the list a day. If items on the list are too big - like wash dishes - break them down into smaller achieveable tasks - like collect all the dirty dishes and stack them next to the sink; then later or another day you could start washing a load of dishes. Or picking up all the clothes on the floor and take to the laundry; then pick out a load of what needs most urgent washing (eg work clothes) to put in the machine. Or if there is something you are avoiding as it needs to be done at a certain time - schedule that task at a time that you can do it - eg. set reminder in phone to call x in lunchbreak. For some people once you get started it can be easier to keep going to completion.
Another strategy that worked for this person was setting a timer. He would set a timer say for every 15mins and start doing the task until the timer went off. When the timer went off he would have a quick break - eg. for a drink. When he first used the strategy he was having 15mins doing tasks, 15mins break, then back for another 15 min task. But over time he would continue the tasks for multiple consecutive periods before needing a break.
When things get out of hand it can be really overwhelming to deal with and can seem like you need to push people away. You could feel embarassed, or not able to cope with how you anticipate they might react to your struggles - potential criticism or them trying to impose their solutions on you, or that they may be overwhelmed with trying to help you. But there are some people who can help if you are willing to give them a go. Perhaps ask people you know if they have struggled with housework in the past - and if others open up about their challenges then you may feel more comortable sharing your challenges and seeking their assistance. Could you collect up your dirty clothes and ask someone to help you wash them - perhaps a paid service if you can afford to, or a friend/family member (you could say your washing machine isn't working properly)?
If you have a problem with your dishes, is this contributing to you not eating properly? If this is you, it may be making your depression worse. Perhaps finding some easy solutions to having proper meals could help you until you can sort through your kitchen. There can be some relatively cheap and nuritious soups available (depending on your budget) or other instant meals depending on your preferences.
Do you know what important letters you are missing? Are there alternative ways to get that information - such as phoning companies who you have bills with and paying what is owing and maybe even setting up a system so the bills get automatically paid when due? Or could you ask for important letters to be emailed to you if this is easier for you to manage?
As you get back on top of your clothes, dishes, letters etc - it can help to set up systems to make it easier to keep on top of those tasks - such as laundry basket/s where clothes accumulate (bedroom/bathrooms); direct debit bills; cooking in batches and freezing meals (to reduce how many dishes you accumulate). It is working out what strategies work for you.
Keep posting on these forums your challenges and achievements as you work through these issues. And others can likely suggest some things to try.