This must be a very emotional time for you when what you really need is some space to grieve and all the usual goings on are just too much to cope with.

Have you told the kids why you need them to behave better atm? It does sometimes help just to say listen, mums not too good atm and it would really make me feel better if you could make more effort to behave for a while. Have a chat with them and see if they respond.

Do you have someone IRL you can talk to, if not maybe seeing your doctor would help, he might prescribe something to help you get through the next few weeks but even if you don't want to take anything just telling someone may help and he should be able to offer you support in other ways. If you think your suffering from depression, don't grit your teeth and bare it, tell your gp and let him treat you. As you have a two year old you could also try contacting homestart (if you google it you'll find there details), I know many people who've used them and they are very supportive. I can't imagine what it feels like to lose a child but I also can't imagine its something you should have to cope with alone.

If you want my advise on dealing with the kids bickering in general, leave them too it! Don't try to sort it out, don't lay blame, just make a rule that if they do it they will both be sent to there rooms and if they continue go deaf. The only time to intervene is if they're REALLY hurting each other, otherwise trying to resolve it will fuel it more as mostly us parents will never really know who did or said what and next thing we know there arguing about getting each other into trouble and off they go again.

Please do go and get some support for yourself though, don't struggle to cope alone when you don't have to.