Here is another option for battery replacement which includes the 5S and 5C.
Note: They are not official apple batteries but do carry a 1 year warranty.
In Store Only YMMV
iPhone Same Day* Battery Replacement $27.99 Reg. $49.99
Supported device (iPhone 5C, 5S, SE, 6, 6 Plus, 6S, 6S Plus, 7, 7 Plus)
While supplies last. Subject to battery availability. Available in select stores.
*SAME DAY CELL PHONE REPAIR
Not available in all stores. Visit officedepot.com/storelocator to find a participating store near you. Available in-store only for select Apple and Samsung products. Place your order by 12:00 PM and request Same Day Service. Subject to parts availability. Limit 5 per customer per day. If order is not complete by the end of regular store hours, your order receives a repair discount of $25.
The following are not eligible for Same Day Service: Devices determined to need additional repair components beyond the screen replacement, devices that may have damage or worn parts due to previous repairs, and devices requiring warranty repair.
Replacement battery is compatible with the following iPhone® devices only: iPhone 5SE, iPhone 5C/5S, iPhone 6S, iPhone 6S Plus, iPhone 6, iPhone 6 Plus, iPhone 7, iPhone 7 Plus. While supplies last. Subject to battery availability. Available in select stores. Visit officedepot.com/stores to find your nearest location and call for availability. See associate for details