I had washed the rims, applied soap water and its not leaking..

 

Shopping for new tires.. not in hurry though..

 

I think trying to patch that nail puncture might be risky - too close to the shoulder and plugging and patching would weaken the tire. As for the cut - you wrote you can see the tire cord ... that makes it easy. Replace the tire! Trying to save save a couple bucks isn't worth the inconvenience of a tire failure on the road OR (WORSE YET) a tire failure that causes an accident.

 

I don't think you need to rush out this minute ... if you aren't making any serious driving trips this week you could get some shopping time under your belt! TireRack has a good inventory of highly recommended tires and reviews so you can let the Internet do your shopping. And many local brick and mortar stores will do price-match too! Check the link in my signature for cash rebates wherever you shop on line too.

 

I would patch the hole, if the tire leaked with the nail removed and not sweat it, but I would also never recommend to someone to drive it like that.

 

Put soapy water on her, if she bubbles she did it!