When will two consecutive months of fasting, which began on Monday, 2/10/1430 AH, be completed?

Since you did not begin fasting from the first of the month, you should fast Shawwal and Dhul-Qa'dah. Dhul-Qa'dah is considered by moon sightings, and Shawwal is considered by number.

So, if Shawwal is complete (30 days), you should fast one day from Dhul-Hijjah. And if Shawwal is incomplete (29 days), you should fast two days from Dhul-Hijjah.

Shaykh Ibn 'Uthaymeen, may Allah have mercy on him, chose to consider Shawwal also by moon sightings, so you would fast only one day from Dhul-Hijjah, whether Shawwal was incomplete or complete, because only one day was missed.

The first opinion (which necessitates fasting one or two days from Dhul-Hijjah based on the number of days in Shawwal) is more cautious and more absolving of responsibility.