Its typical and hylotelphium sedums to have dead shoots at the base- this is how they hibernate in the winter. Your one seems to particularly old and weathered, this mound looks exposed .

This plant is never going to look good again if you just leave it. You can either wait a bit till the shoots are a somewhat bigger, and then dig the clump up, break it apart, and try to get some decent pieces with good roots and very little of the dead stuff, or you can simply divide the other plant. In time these plants tend to get too big and kind of fall apart in the centre, with the stems falling in all directions. So it would also benefit the big one. If you want more plants, you can always use the little pieces, and let them grow on somewhere else.