Wells, 26, has previously played for Belper Town, Basford United, Stamford and Hednesford Town.
"I'm absolutely delighted to have signed Kieran," said manager Martin Carruthers.
"Mark and I have worked with him in the past at Basford and he has always scored goals at every level he has played at.
"His energy, enthusiasm and personality will ensure he becomes a real crowd pleaser and fans favourite I'm sure. I have followed him closely over the past few seasons and he has developed and improved as a player. He will become a real asset to the club."
Christian Oxlade-Chamberlain, younger brother of England international Alex Oxlade-Chamberlain, also recently joined the club on an initial one-montj loan deal from National League side Notts County.
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