So I feel like I have to teach her to sleep independently but have no idea how
To go from where you are now to independent sleep is going to be either (a) a very long process or (b) brutal and involve a lot of distress.
Babies don't have to be independent sleepers to be good sleepers.
A family who do planned cosleeping, family bed, all go to bed together and offer on-demand feeding to sleep and affection from the beginning of the night through to the end of the night may well all get loads of sleep. Mum and Dad being well rested from early nights, no one having to get up or out of bed at any point, sharing the load with cuddle reassurance (both for baby and for each other) and fast, effective resettles for baby with minimal disruption to anyones sleep.
You seem to be comparing yourself to families who opt for early established independent sleep. That's fine, its a great option to get yourself a good sleeper. But its not the only way.
Your current parenting style seems to be far more aligned with the family bed that it does with independent sleep. My thoughts (for what they are worth) is that most sleep battles arise when parents want to swap from the two opposites - so they have an attachment-type parented baby but want independent-sleep type outcomes. I think if you want to establish gentle (ie no crying) independent sleep it needs to be done early, by 2 months old or so. Certainly in time for the 4 month regression. After this time it's going to be brutal and involve distress in one kind or another.
If you just accept that your way is of the feeding to sleep and cosleeping variety, there need be no battle and you could all get yourself more sleep.
And if you do decide you want/need baby to sleep independently, then there are kinder ways to do it than controlled crying. Have a look at the opening post of this thread for one idea, based on speeded up Gradual Withdrawal:
www.mumsnet.com/Talk/sleep/1394888-What-worked-for-us-Hope-this-helps
So I would set up the cot as a sidecar (if you haven't already) so baby has that extra space if needed, but