The sum of some parts
05AB1E, 10 bytes

æʒø`O²‹_i,


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Explanation

æ           # compute powerset of input
 ʒ          # filter out falsy values (we don't actually filter, it just saves a byte)
  ø         # zip into a list of keys and a list of values
   `        # push separately to the stack
    O       # sum values
     ²‹_i   # if not less than input_2
         ,  # print list of keys