Numbers – continuous line connecting gapped data

You can use the VALUE() function for that. Its only argument is the cell that you want to evaluate

From the documentation:

The VALUE function returns a number value, even if the argument is formatted as text. This function is included for compatibility with tables imported from other spreadsheet apps.

VALUE(source-string)
source-string: Any value.

Notes
You’ll never need to use the VALUE function in a new table, because numbers in text are automatically converted for you.

Only the formatted text is converted. For example, if you type the string $100.001 into a cell, the default format will display only two decimals ($100.00). If VALUE refers to this cell, it will return 100, the value of the formatted text, not 100.001.

If the argument can’t be returned as a number value (does not contain a number), the function returns an error.

Examples
=VALUE(“22”) returns the number 22.

=VALUE(RIGHT(“The year 1953”, 2)) returns the number 53.

As far as I know there is no way to make the desired changes all at once, but there is a way that you can make things significantly faster than creating a brand new chart.

As it appears you've already done, copy and paste the chart into your Numbers worksheet. Depending on the chart you've created, select an individual line, bar, pie segment etc. and open the chart pane in the inspector. Here's a screenshot of what that looks like (with "Example 2" being the copied and pasted chart):

As you can see, in the field labeled "Data," Table 1 is referenced. This source can easily be changed to a different table. Just delete the listed source and select the new source of data from your worksheet (you don't have to type in the source). Depending on your chart you may also want to change the data source for the x-axis labels. This is also controlled by the inspector. That can be changed here:

While you'll have to change the data source for each line of the chart, it's certainly faster than creating a whole new chart and having to format it the way you'd like.