Is there an easy fix for an old iPhone 5s battery? I have had my iPhone 5s for almost two years now, and the battery has been dying quickly. For example, even when my phone is on airplane mode, it will drop from 90% charged to 85% charged within two hours. I'm pretty sure this is due to the battery life.

Is there an easy fix that will not cost me more than replacing the phone?
Battery Drop of 5% in two hours is not bad

A battery drop of 5% within two hours is not bad at all. How much does it drop throughout an entire day? How long can you use it for taking a few calls, browse a little?
If it only drops heavily, when not in airplane mode, please check your battery usage: Settings -&gt; General -&gt; Usage -&gt; Battery Usage

Replacement

Apple offers an about $80 (+ $7 shipping) battery replacement-program for iPhones https://www.apple.com/batteries/service-and-recycling/

Alternatively, if you - or a friend of you - like to fix things yourself you could consider a iFixit iPhone 5s Battery replacement kit, which is about $30 (without shipping)  https://www.ifixit.com/Store/iPhone/iPhone-5s-ReplacementBattery/IF124-002-2

iFixit provides very useful guides of how to fix electronic devices from Apple or other manufacturers.

Check warranty

You might want to also check your warranty status: https://selfsolve.apple.com/agreementWarrantyDynamic.do

If you have an AppleCare Protection Plan they might replace the battery for free!