Correction on My Above Response.

After I studied the H-1/L-1 new rule which was effective on Jan. 17, 2017, I found that this rule just applies to the person who has H-1/L-1 (including H-4/L-2 dependent) with unexpired I-94, and if he is laid off before the I-94 expires (at least the I-94 has the duration of  60 days after layoff), then he has the grace period of 60 days. But if his H-1/L-1 I-94 expires, then he has the grace period of only 10 days. So, you do not have 60 day grace period, but have only 10 days after your L-2 I-94 expires. So, you have to file I-539 before the L-2 I-94 expires and thus, you can legally stay in US.