The time the singer had for singing was 80 - 10 = <<80-10=70>>70 minutes.
The time the singer had for singing 5-minute songs was 70 - 10 = 60 minutes.
The number of 5-minute songs was 60 / 5 = <<60/5=12>>12 songs.
The total number of songs sung is 12 + 1 = <<12+1=13>>13 songs.
#### 13