Let's start by looking at the digests. First the lamba phage is cut with
-EcoRI to yield 20kb, 13kb, 12kb, 5kb (total is 50kb)
-BamHI 22kb,14kb, 10kb, 4kb (total 50kb)
-BamHI and EcoRI to yield 14kb, 13kb, 9kb, 6kb, 4kb, 3kb, 1kb (total is 50kb)

Reconstructing the plasmid is kind of like putting together a puzzle. Let's start with trying to identify bands:
-both of the large kb bands are cut in the double digest which means they contain both restriction enzymes
-the 14kb in the double digest may be the BamHI-14k-BamHI or it might have been a cleavage product of the
-likewise with the 13kb fragment
-the 9kb double digest has to be a BamHI-9kb-EcoRI fragment because it is a new fragment
-the 6kb double digest has to be a BamHI-6kb-EcoRI fragment because it is a new fragment
-the 4kb may be a BamHI-4kb-BamHI or a BamHI-4kb-EcoRI fragment
-the 3kb is a BamHI-3kb-EcoRI
-the 1kb is a BamHI-1kb-EcoRI

-BamHI-3kb-EcoRI-1kb-BamHI-4kb-EcoRI-6kb-BamHI-14kb-BamHI-EcoRI-13kb-EcoRI-9kb-BamHI-

In order to determine the order of the fragments all you need to do is try to reconstruct the fragments that are shortened even more in the double digest. It's a matter of trial and error. You should try it.

No let's look at what we know of the insert TN-10. It is 9kb long with 1 EcoRI site 3.5kb from the left (TN-3.5kbEcoRI-5.5kb-TN) with no BamHi sites. The digest of the vector+insert yields:
-EcoRI 20kb,13kb,12kb,9.5kb,4.5kb
-BamHI 22kb,19kb,14kb,4kb
-EcoRI and BamHI 14kb,13kb,9kb,8.5kb,6kb,4.5kb,3kb,1kb

If we look at the single digests we can see that nearly all of the BamHI fragmetns are the same prior to TN-10 insertion except the appearance of a 19kb fragment and the disappearance of the 10kb fragment. Therefore the TN-10 inserted into the 10kb fragment of BamHI. For the orientation we must now turn tothe EcoRI digest. Comparing the single digest pre and post TN-10 insertion the major differences include:
-the 5kb fragment is gone post insertion and replaced by a 9.5kb and a 4.5kb fragment.

Now look at the lambda vector map. We have two places to put the TN-10 vector; in the BamHI-4kb-EcoRI or the EcoRI-6kb-BamHI fragments. Now because we see the 6kb fragment in the EcoRI and BamHI double digest it must be in the 4kb fragment. To determine where in that fragment we look at the remaining fragments we need to account for (we will use the EcoRI digest as it is the simplest; EcoRI-4.5kb-EcoRI and EcoRI-9.5kb-EcoRI). In order to make the 4.5 fragment from the available sites we must place the TN-10 fragment insertion site right at the BamHI cut site in that area. We have now established the direction of the fragment.