God created the angels even before the world existed. They are spiritual beings (although sometimes they can take visible form) and were created to be God's servants and messengers.
Although the Bible doesn't answer all our questions about angels, it does tell us that God has given them the task of protecting and helping His children. The Bible says, "Are not all angels ministering spirits sent to serve those who will inherit salvation?" (Hebrews 1:14). Elsewhere the Bible says, "For he will command his angels concerning you to guard you in all your ways" (Psalm 91:11). This is God's promise to you.
Although we do not see them, I believe we are surrounded and protected by God's angels every day. Does that mean we will never experience accidents or illness? No, it doesn't as you have discovered. But it does mean Satan will never be able to snatch us away from Christ if we belong to Him, and even at the moment of death the angels will take us safely to heaven.
Let me ask you a question: Do you know Christ? If not, perhaps God delivered you from this accident so you would stop and realize your need of Him. Don't ignore His warning, but face your need of Christ and open your heart to Him. It is the most important decision you will ever make.