Have you tried cleaning the dock connector on the bottom of the phone? (Also consider checking it for damage with a light & magnifying glass.)

Make sure the top "sleep/wake" button isn't sticking. You may want to try replacing the battery. *Buy one that comes with a "pentalobe" screwdriver. While you wait for your battery to come in the mail, you can take the following steps: 1) Power the phone OFF. 2) Clean the the dock connector-socket on the bottom of the phone with an old toothbrush. Get all lint/dirt out of there. Wet your toothbrush with some 91% rubbing alcohol from WalMart and gently scrub the connector. Make sure there are no bent/corroded pins. You can dry it out with a hairdryer for a few seconds. 3) Look into the dock connector closely with a flashlight. You will see a small rectangular hole. Inside the hole there is a sticker. Is it red, pink or mostly white/yellow? 4) Now look into the headphone jack on top of the phone closely with a flashlight. The small dot in the bottom, is it red, pink, or mostly white/yellow? *If you find anything other than white/yellow on the stickers in the dock connector or headphone jack, your phone MAY...

What exactly happened to the iPhone while in your daughter's possession? You may possibly be experiencing a failure in the logic board "power management ic", assuming your replacement battery-cable-connector wasn't somehow defective. It is generally an uncommon occurrence (and an expensive fix, unfortunately), but some experts think cheap, "fake" Apple chargers/cables are to blame for the bulk of these failures. Please check out this article to see if it sounds like your problem: https://www.mendmyi.com/blog/2014/06/has-your-iphone-5-died-and-wont-charge-or-come-back-on-we-can-fix-that/

I get by fine with a "Nulink" 2-pc screwdriver set ($7) and a "Delcast" Kit ($10) I purchased on eBay. The Nulink Philips/Pentalobe set should be adequate for most screws you will encounter.

Do you have a loupe or microscope? Are the LCI's pink/red? Possible corrosion, bent pin(s), or damaged digitizer connector. Does the LCD-Digi work fine when connected but not pressed into phone's frame? Possible excessive pressure being applied to the LCD, due to bent/distorted frame. That's all I've got, with limited info.

I've refurbished more than a handful of LCD's and I use a special "LOCA Remover" and a plastic razor blade that you can pick up on eBay. The remover works quite well, but unfortunately it's only useful after glass has been cut off using a "hot plate/separator tool". In my opinion, this is where the most risk is, especially on the previously refurbished LCD's, which were bonded with LOCA. I'm sorry to report my success rate on these LOCA units are abysmal. You've got to get your hot plate up quite hot (100C max) to soften up the LOCA enough to cut thru it. And it will not soften up anything like the dry sheet adhesive. When you have an LCD with a spider web of long shards of broken glass, it can be a nightmare releasing the glass from the LOCA, without cracking the digitizer underneath. I hope you have better luck than me! The new LCD's or LCD's which were bonded with the dry sheet adhesive you mentioned (the RIGHT WAY to do it), are by comparison a cinch! ***When the dry OCA sheets are used, you also...

If your replacement lcd-digitizer assembly was refurbished, the digitizer could have been stressed to make it failure prone. LCD's are easily damaged, depending on the refurb and bonding process. Liquid (LOCA) glue residue found near/on flex cables is often a hint. Also, another good possibility involves the flex cable connectors themselves. Did the replacement LCD-digitizer cables have foam pads on the back side of the connectors or just shiny metal? If those foam pads aren't there, when you go to place the connector's cover with the 3 screws, any one of cables have room to pop off when positioning the display. When you press down on the display, it may be enough to flex the display and temporarily or partially make a connection at the cable ends (upper right corner of screen) underneath and/or nudge a defect (crack) present in the digitizer panel, which is just below the glass surface.

For the sake of your phone please STOP trying to turn it on! At a minimum, you will need to get a new battery. The battery is the 1st thing to get ruined or short out when wet. When that is replaced, it will most likely turn on, but you are not out of the woods... Your iPhone's days will be numbered but a few, depending on the extra steps you take. Just because it turns on now doesn't mean it won't crap out a few weeks down the road... Resist the urge to turn it on for now. You (or someone qualified) should remove the back cover (2 bottom special screws) and disconnect the battery to prevent damaging corrosion on your circuit board. You need remove battery power and your circuit board needs to be cleaned in an IPA (alcohol) bath. If you're willing to take an extra step to protect your iPhone, you should get a professional "Ultrasonic Cleaning" and a new battery. If corrosion found on the logic board is minimum, many potential problems are eliminated. Sometimes, peripherals such as cameras can also start...

The phone likely needs a complete disassembly and the logic board deep cleaned using an IPA (alcohol) bath at a minimum or an ultrasonic bath in distilled water (best). *Impurities/minerals present in the water get stuck on the electronics and cause corrosion. This should be removed ASAP if you're interested in saving your device. As you continue using it after liquid exposure, corrosion gets progressively worse until the device loses power due to the increased electrical resistance. Eventually, corrosion is built up to a point where device cannot be repaired.

Does the phone respond with a vibrate (loop?) when connected to charge? Does the phone respond with a chirp when connected to iTunes via USB? You may want to try a "Reset" or "Restore". Reset: https://support.apple.com/en-us/HT201559 Restore: https://support.apple.com/en-us/HT201263 *Make sure data is backed-up first! If Display will still not turn on, it is possible a connector inside the phone may be loose or the LCD backlight may not be turning on. *Look very closely at the screen with a flashlight. **A small percentage of iPhone 5/5S's have also unfortunately experienced some "Power Management IC" failures, where it suddenly fails to charge the battery. It may "pretend" to charge or give a false reading. This is a possibility that can be looked into after ALL other simple problems have been eliminated, as this is a logic board chip failure.