Alignment is very important. Not only do you have the battery connections, but the small contacts for the ‘gas gauge’ in the same connector. Align the connector to the main board carefully. Then press down gently but firmly and very slightly rock the battery connector as you press down. When the connector seats it will make a ‘click’ sound. Then press down gently and make sure both sides are seated down and locked in place.
If the phone still won’t boot when prompted, there are many, many issues that could be involved that prevent the phone from booting. One of the main reasons is from using non-Apple certified cables and chargers with the phone. These gas station cables and chargers that are not Apple certified can damage the charging IC on the main board. This chip is responsible for making sure the battery isn’t overcharged or overheated during charging. If it is damaged, the phone won’t boot.
iPadRehab can give you a quote to repair the phone. You should contact them at their web site and get a quote. And never use gas station cables or chargers with your iPhone. Eventually, there could be damage that takes an expert to repair.