Mindfulness as Heartfulness

Jon Kabat Zinn explains that Mindfulness in Asia means 'Heartfulness' He describes integrating buddhist tenets such as beginner's mind, non-judgment, letting go, acceptance, trust, gratitude and generosity into mindfulness as part of informal meditation. My difficulty with my formal and informal meditation practice through the course of the day seems mechanical because I don't know how to cultivate and integrate heart into it. I am aware of my senses. I have the mind be still with little or no thought. But there doesn't seem to be anything warm, understanding, loving or appreciative. It doesn't feel rewarding at all beyond discipline of the mind. I ask myself, if mind must be still and thoughts silent, how to I cultivate matters of the heart in my meditation. Or do I allows for cultivating a 'Heartful' narrative that explores the buddhist tenets mentioned above. It all just feels like cold discipline to me at the moment and nothing more. Please help me understand what mindfulness may look like in the context of heartfulness.