Generating random floats, summing to 1, with minimum value

I saw a many solutions for generating random floats within a specific range (like this) which actually helps me, and solutions for generating random floats summing to 1 (like this), and separately solutions work perfectly, but I can't figure how to merge them.

Currently my code is:

import random
def sample_floats(low, high, k=1):
    """ Return a k-length list of unique random floats
        in the range of low <= x <= high
    """
    result = []
    seen = set()
    for i in range(k):
        x = random.uniform(low, high)
        while x in seen:
            x = random.uniform(low, high)
        seen.add(x)
        result.append(x)
    return result

And still, applying

weights = sample_floats(0.055, 1.0, 11)
weights /= np.sum(weights)

Returns weights array, in which there are some floats less that 0.055

Should I somehow implement np.random.dirichlet in function above, or it should be built on the basis of np.random.dirichlet and then implement condition > 0.055? Can't figure any solution.

Thank you in advice!