For this kind of issue, please follow up the following steps to check:
1.confirm the battery and ac adapter connect the laptop normally.Maybe the bad connect lead to this problem. If there are no lights when you turn on laptop, one of possible reason is that your adapter is malfunction.
2.the second possible cause is laptop dc jack failure. If the dc jack is broken or loose, you are required to replace it or repair it.
3.Another possible reason is that there is something wrong with laptop motherboard. For this situation, it includs two kinds of possibility,a failure in an electronic circuit and BGA malfunction.
For more detailed reasons and solutions, please have a look at the following article:http://www.ifitjams.com/lappower.htm
If you verify your laptop adapter is dead and you plan to order a new one, I recommend one for you on the basis of your laptop model:Replacement Toshiba Satellite L655 AC Adapter 65Watt 19V 3.42AReplacement Toshiba Satellite L655 AC Adapter 75Watt 19V 3.95A