:( sorry things are so hard.
Are you absolutely sure, if you reduced your workload, you wouldn't be ok? Say, one of you quit/reduced hours, and therefore had less childcare costs? Maybe you could do some tutoring instead?
Are you sure you are on the best repayment plan for your debt? And are you definitely getting everything you are entitled to - childcare vouchers, tax credits maybe? CAB can help with both of these issues, they can do a full calculation including hypotheticals about quitting work etc.
Have you looked into getting a childminder instead of nursery? They could look after your youngest and then pick up the older from school?
Are you both at the same school - do you definitely need both cars? Could one of you drop the other off early before going to their school? Is there a colleague you could carpool with? Do you get a petrol allowance from work? Any chance you could cycle sometimes?
What about one or both of you stepping down from the HOD role? Horrible I know, and less money, but presumably less work too. Do your colleagues/superiors know how hard it is for you, could you delegate a bit? Are you entitled to time off and sick pay due to your illness?
I'm guessing you don't have room for a defined study space. Any way you can make one? A garden shed? Or is there a library you can get to? I've found (admittedly only with studying rather than actual work) that I get a lot more done - even factoring in travel time - when I'm away from the distractions of home. Maybe you could take it in turns to have a few hours away to get more work done?
The dogs... I can't imagine how hard it must be but I don't see how you can keep them. There must be rescue centres around, I've visited some lovely ones where absolutely no healthy animal is put down. Maybe they would let you visit. They must cost you so much (do you have pet insurance?) but it seems they don't get as much attention as you'd like to give them, so in a rescue place they may be better off.
How many clubs do the DCs do? They must be exhausted - that could be part of the reason for playing up. If they miss you they will delay bedtime, negative attention is better than none to children. The clubs must cost a lot too. Can you reduce to one activity per week each?
How are your utility bills? Are you on the cheapest provider? Some companies give away a free monitor. Bit late now but we got some amazing film stuff which you put over the window frames to act like an extra 'glaze' - saved an astonishing amount on heating. Got hippo bags in the toilet? Water butt outside if you have plants? DCs sharing a bath or just having quick showers? Do you have expensive tv channels etc? I'm sure they will not miss cbeebies if they have a few DVDs. We recently disconnected our tv altogether and DD has been fine. Can you get to charity shops for clothes, or swap with friends?
What about food shopping - any way you can switch to somewhere cheaper, or to 'lower' brands? Do you buy in bulk? Do you get things like frozen veg and tinned pulses which are cheap and bulk out meals? Can you set aside a couple of hours now and again to cook massive amounts of stew etc and freeze them? Do you shop online? It costs for delivery but saves so much time especially if you save a list of all your usual items.
It means spending more money but is it at all feasible to outsource some housework? A cheap cleaner or a laundry service? Or you may just need an iron tight schedule. Say, every night as soon as the DCs are asleep you both do half an hour cleaning before anything else. Then have a 2 hour blitz every weekend or something - all 4 of you. The little ones can help with collecting laundry, tidying their room and so on. Maybe for dealing with DCs and doing your work you could take alternate days, and put up a chart so you both know what you need to do each day.
Sleep is a damn difficult one. Have you tried a reward chart? Maybe you could tie it in with weekend activities if they sleep through, or a £3 DVD or something? Have you watched any Supernanny, I know she's controversial but her 'rapid return' technique seems to work well after a few traumatic nights. Have you checked they aren't scared or something, do they have a nightlight? A magic wand to wave the monsters away? (tried and tested for my DD...) I am absolutely sympathetic to how easy it is to get into a mess with this kind of thing, but you just can't carry on like that, so you might need to push through a big change even though it's hard at first.
Also do you have a lot of clutter? Old clothes, toys etc? Maybe in summer you could do a boot fair or even a yard sale at your own house. Be ruthless about what you're getting rid of, and anything not sellable can be recycled. I have only just started doing this but it's made an amazing difference to our tiny house.
Sorry, this is quite possibly my longest ever post and certainly the one with the most question marks
but I hope it's given you some ideas for changes. It is quite probably circumstances that drove you to this, and with time sorting all this out could help you fall in love all over again and have a better life.