Husband Not Comforting Newborn Baby

I have a wonderful husband who works hard every day hauling equipment and I am a stay at home mom at the moment. We have a five week old daughter that is breastfed. My daughter refuses to sleep anywhere but her swing or on my chest while I'm sitting in my recliner, and therefore I've only been averaging two or three hours of sleep per day since she was born. I realize for a lot of mom's they have gotten to the point where that is more than enough, but I am exhausted when my husband comes home and still can't take a nap because once I leave the room she will cry because I'm not there and he won't pick her up. He will just keep watching TV and let her cry and then be mad at me when I get back up to comfort her after 20 minutes of crying. I can't let my baby just "cry it out" as he suggests because when I have tried it a couple times she seems so sad and distant afterwards and it breaks my heart. Swaddling does not work to comfort her, it only makes her angrier.

He never plays with her. He will occasionally feed her and change diapers, but I wish he would try to establish more of a bond. I need advice!