These are truly the hardest days of caring for an infant. Exhaustion is real and you still want to give your 5 year old time...hard to take that "sleep when the baby sleeps" advice in every baby book.
Nows the time to call in the reinforcements...ask Dad to take a vacation or personal day to give you a chance to sleep while the baby sleeps for a full 24 hours.
Your 5 year old could probably use this bonding time with Dad and it won't hurt Dad either. When he sees that you are literally shaking with sleep deprivation(I'm assuming...I sure know what it feels like) he'd probably be happy to be a hero and save you.
Then call in reinforcements...a relative, a baby sitter, a cleaning service(that always contributes because you try to keep order when baby sleeps instead of sleeping too)...who can love on your 5 year old or help around the house.
You just have to make sleep happen, even if its just a couple hours at a time throughout the day.
Remember, this is temporary. Your 5 year old will be o.k., and your infant will sleep through the night one day and you will be refreshed again...Remember things like housework, can be farmed out (and help boost your spirit when done), put a stack of books near your bed and read to your 5 year old (or let them "read" to self) take a walk with baby in buggy in the morning (even though hot) to get some sunlight and let the 5 year old play in a park...it will help those blues just by getting some sun...
For now, don't feel guilty about needing sleep during the day, just lie in bed and nurse baby...Sending God's richest blessings your way!