The quick and easy solution is to have someone open your phone for you, disconnect the battery, and reconnect it. This will solve 1970 right away and preserve your data. This is a job that really any independent repair shop can do for you, but be aware that opening your phone by anyone can potentially disqualify you from a future Out of Warranty swap at Apple--which means turn in your phone, pay $269 for a 5s, and get a refurbished one. The current value of a good condition 5s is much less than $269, so I would recommend you just pop down to your local independent repair shop and get 'er done. This is a 5 minute or less job with 4 screws.
Doing it yourself is not recommended--the 5s touch ID sensor is easily torn when opening the phone without experience.