i dropped my iphone 5 and it wont turn on now. it keeps cycling from the dead battery to the apple when plugged in.

If your iPod Touch, iPhone, or iPad is Broken

Apple does not fix iDevices. Instead, they exchange yours for a refurbished or new replacement depending upon the age of your device and refurbished inventories. On rare occasions when there are no longer refurbished units for your older model, they may replace it with the next newer model.

ATTN: Beginning July 2013 Apple Stores are now equipped to do screen repairs/replacements in-house on iPhone 5 and 5C. In some cases while you wait. According to Apple this is the beginning of equipping Apple Stores with the resources needed to do most repairs for iPhones, iPads, and iPod Touches that would not require major replacements. Later in the year the services may be extended as Apple Stores become equipped and staffed with the proper repair expertise. So, if you need a screen repaired or a broken screen replaced or have your stuck Home button fixed, call your local Apple Store to see if they are now doing these in-house.

You may take your device to an Apple retailer for help or you may call Customer Service and arrange to send your device to Apple:

Apple Store Customer Service at 1-800-676-2775 or visit online Help for more information.

To contact product and tech support: Contacting Apple for support and service - this includes international calling numbers.

You will find respective repair costs in the appropriate link:

iPod Service Support and Costs

iPhone Service Support and Costs

iPad Service Support and Costs

By the sounds of the symptoms after the drop, that would most likely be your only option.

Sorry

Pete